Regularities of rings

REGULARITIES OF RINGS

C. ROOS

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REGULARITIES OF RINGS

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. B. BOEREMA, HOOGLERAAR IN DE AFDELING DER ELEKTRO-TECHNIEK, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN, TE VER-DEDIGEN OP WOENSDAG 22 JANUARI 1975 TE

14.00 UUR t J ISJ CO - J »-• • ^ OP DOOR

CORNELIS ROOS <''^^^^JP^\

geboren te 's-Gravenhage V?&, DOtltNSIH.lOr

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Dit proefschrift is goedgekeurd door de promotor PROF. DR. F. LOONSTRA

Hoe moeten wij de wetenschappen, zoals de geneeskunde, derechtsgeleerdheid, de ster-renkunde, de wiskunde en de wijsbegeerte beoordelen? Zullen wij niet opmerken dat zij gaven van God zijn die in de geesten der mensen als het ware ingedruppeld worden? Zullen wij niet in de wetenschappen de goed-heid van God zien en erkennen; opdat Hij in de kleinste en in de grootste dingen de lof en de eer ontvange?

(Enigszins vrij naar het commentaar op Jes. 28 :29 van JOHANNES CALVIJN).

Aan mijn ouders Aan Gerda

Gaarne betuig ik mijn dank aan de directie van de firma UNION OIL, gevestigd in Lisse, die deze wijze van vermenigvuldiging van dit proefschrift mogelijk maakte.

CONTENTS

page

INTRODUCTION 7 CHAPTER 0: SOME RADICAL THEORETICAL RESULTS 9

CHAPTER I: A GENERAL TYPE OF REGULARITY FOR

RINGS 13 1. Definitions 13 2. The r a d i c a l and the subradical determined by a regularity 13

3. Coincidence of the radical and the subradical 16

4. Equivalence of r e g u l a r i t i e s 17 5. A survey of well-known r e g u l a r i t i e s 18

CHAPTER II: POLYNOMIAL REGULARITIES 21

1. Introduction 21 2. The main theorem; p - r e g u l a r i t i e s 21

3. Summability of r e g u l a r i t i e s ; s p - and p ' - r e g u l a r i t i e s 24 CHAPTER HI: OVERNILPOTENT AND UNDERIDEMPOTENT

REGULARITIES 28 1. Overnilpotent r e g u l a r i t i e s 28 1. 1. Overnilpotent p - r e g u l a r i t i e s 28 1.2. Overnilpotent s p - r e g u l a r i t i e s 29 2. Underidempotent r e g u l a r i t i e s 31 2. 1. Underidempotent p - r e g u l a r i t i e s 31 2 . 2 . Underidempotent s p - r e g u l a r i t i e s 32 CHAPTER IV: UNIC POLYNOMIAL REGULARITIES 33

1. P r e l i m i n a r y r e m a r k s 33 2. An intersection theorem for unic p - r e g u l a r i t i e s 34

3. An intersection theorem for unic p ' - r e g u l a r i t i e s 36

4. The r e g u l a r i t y JRf}, f unic 37 5. Tiie regularity |<Rf >}, f unic 45 6. The regularity JRfR}, f unic 49 CHAPTER V: NULLIC POLYNOMIAL REGULARITIES 51

1. An intersection theorem for nullic p - r e g u l a r i t i e s 51 2. An intersection theorem for nullic p ' - r e g u l a r i t i e s 52

3. The regularity {Rg}, g nullic 53 4. The regularity {<Rg>}, g nullic 56 5. The regularity {RgR}, g nullic 63

CHAPTER VI: MONOMIAL REGULARITIES 65 1. Introduction 65 2. Unic m - r e g u l a r i t i e s 65 3. Nullic m ' - r e g u l a r i t i e s 66 4. Nullic m - r e g u l a r i t i e s 73 REFERENCES 85 SUMMARY 89 SAMENVATTING 90 KORT LEVENSBERICHT VAN DE PROMOVENDUS 91

INTRODUCTION

The purpose of this thesis is to develop a general theory of r e g u l a r -ities for rings. In ring theory many so-called r e g u l a r i t i e s appear. The oldest one s e e m s to be the von Neumann regularity. In 1936 VON NEUMANN [36] defined a ring R (with identity) to be regular if and only if for any element a of R there exists an element x of R such that a = axa. In 1950 BROWN and McCOY generalized to rings without identity, and they succeeded in proving that any ring contains a g r e a t -est regular ideal [15], In the meantime (1942) PERLIS had introduced the concept of quasi-regularity for algebras with identity [38]. He defined an algebra A with identity to be q u a s i - r e g u l a r if and only if for any element x of A there exists an element y such that x + y + xy = = 0. In 1945 JACOBSON [2 7] generalized this concept to a r b i t r a r y rings without identity, and he showed that any ring R contains a g r e a t e s t q u a s i - r e g u l a r ideal, called later on the Jacobson radical of R. It is an easy thing to extend this list of r e g u l a r i t i e s . In section 5 of chapter I m o r e examples are given. In 1950 BROWN and McCOY attempted to define a general concept of regularity for rings [14]. At that time their theory was general enough. All r e g u l a r i t i e s introduc-ed up to then were regularities in the sense of it. However, in 1971 a wide class of regularities was introduced by GOULDING and ORTIZ [25], MCKNIGHT and MUSSER [34], MUSSER [35] and ORTIZ [37], namely the so-called (p,q)-regularities, and it was noted in [25] that some of these (p,q)-regularities fail to satisfy the set of axioms of the theory of BROWN and McCOY.

After these introductory historical r e m a r k s it will be clear that there i s need of a general concept of regularity for rings which includes all r e g u l a r i t i e s known up till now. Our aim i s to supply in this need. The s e t - u p of our thesis is as follows.

In chapter 0 some radical theoretical r e s u l t s are enumerated. Especially the notion of ' s u b r a d i c a l ' introduced in this chapter will prove to be of great importance for our goal.

In chapter I a general type of regularity for rings is defined, and it is shown that any regularity F - r e g u l a r i t i e s will be denoted by italicized capitals - determines a subradical F and a radical F . Sonie n e c e s s a r y and sufficient conditions for coincidence of the subradical F and the radical F a r e given. An equivalence relation in the c l a s s of all regu-l a r i t i e s is defined in an obvious way. Finaregu-lregu-ly, aregu-lregu-l known r e g u regu-l a r i t i e s a r e listed and it is established that each of them is a regularity in the s e n s e of our theory.

In chapter II we develop a method to generate r e g u l a r i t i e s with the help of a finite set of integral polynomials. The arising r e g u l a r i t i e s a r e called polynomial r e g u l a r i t i e s , or monomial r e g u l a r i t i e s if each of the polynomials concerned is a monomial. All r e g u l a r i t i e s Imown up till now in ring theory a r e polynomial r e g u l a r i t i e s , as we shall s e e .

In chapter III the notions of 'overnilpotence' and 'underidempotence' for a regularity a r e defined, and n e c e s s a r y and sufficient conditions a r e derived for a polynomial regularity to be overnilpotent of under-idempotent.

The polynomials used in describing the known regularities a r e all unic (i.e. have constant t e r m unity) or nullic (i.e. have constant t e r m z e r o and lowest degree coefficient unity). A polynomical regularity i s called unic if all the polynomials concerned a r e unic, and nullic if at least one of them is nullic and the remaining are unic.

In chapter IV it i s shown that any unic polynomial regularity can be thought of as the intersection of so-called elementary unic polynom-ial r e g u l a r i t i e s and a unic monompolynom-ial regularity. Roughly spoken, a polynomial regularity is called elementary if all but one of the poly-nomials concerned equal the constant polynomial unity. The element-ary unic polynomial regularities a r e investigated separately; for each of them s t r u c t u r e theorems for the corresponding subradical a r e derived.

In chapter V we obtain analogous r e s u l t s for nullic polynomial regul a r i t i e s . Any nuregulregulic poregulynomiaregul reguregularity proves to be the i n t e r s e c -tion of some elementary polynomial r e g u l a r i t i e s , each of them being either unic or nullic, and a nullic monomial regularity. The element-ary nullic polynomial regularities a r e studied separately again. Finally, chapter VI is devoted to monomial regularities which a r e either unic or nullic. The unic monomial r e g u l a r i t i e s will appear to be r a t h e r uninteresting. Concerning the nullic monomial r e g u l a r i t i e s we show that, up to equivalence, t h e r e exist precisely thirteen of them, eight of which a r e known r e g u l a r i t i e s .

C H A P T E R 0

SOME RADICAL THEORETICAL RESULTS

In the early 1950's both KUROSH [31] and AMITSUR [1], [2], [3] s t a r t e d the development of a general theory of radicals for r i n g s . Up till now many o t h e r s contributed to the further development of that theory. In 1965 DIVINSKY [19] published an excellent survey of the results obtained then. In the next we give a brief s u m m a r y of that part of the theory of radicals which is needed in our study of regularities for rings. Concerning the known r e s u l t s obtained before 1965 we s h a l l base ourselves mainly on the book of DIVINS-KY. Let us note before that in this t h e s i s the rings a r e assumed to be associative and that they may fail to have an identity. Let p denote a (nonempty) class of rings. The c l a s s p is called a

radical class of rings if it satisfies the following t h r e e conditions

[19, page 3]:

(A) p is homomorphically closed, i. e. any homomorphic image of a ring belonging to p belongs to p;

(B) For any r i n g R the set of ideals of R belonging to p contains a greatest element, denoted by p(R) and called the p - r a d i c a l of R;

(C) For any r i n g R: p(R/p(R)) = 0.

If 0 i s a radical c l a s s of rings then, as a consequence of this defini-tion, p(R) is the g r e a t e s t ideal of R belonging to p for any rjng R. At' the s a m e time p(R) is the smallest ideal of R such that R/p(R) has z e r o p - r a d i c a L F u r t h e r m o r e , let p denote the function assigning to

every ring R its p - r a d i c a l p(R), i.e. p(R) = p(R).

Obviously the function p satisfies the conditions (A'), (B'), (B^^) and ( C ) .

(A')If a : R - S is a ring homomorphism, then 0!(p(R)) c p ( a ( R ) ) ; (B')lf A is an ideal of R such that p(A) = A, then A c p(R); (Bi)p(p(R)) = P(R) for any ring R, i. e. p is idempotent; ( C ) p(R/p(R)) = 0 for any ring R.

Now let p be an a r b i t r a r y function assigning to every ring R an ideal p(R) of R. Then, if p satisfies the conditions (A'), (B'), (B^) and ( C ) , p will be called a radical fmiction, or shortly a radical. If p merely s a t i s f i e s the conditions (A'), (B') and ( C ) , then p will be called a

subradical. Hence, according to this definitions, a subradical p is a

r a d i c a l if and only if p is idempotent. In the future we shall meet much subradicals which a r e not idempotent. At this time we give a s i m p l e example of this phenomenon. Define p(R) = R 2 for any ring R. Then p evidently satisfies the conditions (A'), (B') and ( C ) . Hence jo i s a subradical. For any ring R we have p(p(R)) = P ( R 2 ) = R 4 . By

taking R = 2Z we obtain that R 2 = 4Z and R 4 = 16Z. T h i s shows that the subradical p i s not idempotent, whence p i s not a r a d i c a l . The next theorem shows that any subradical gives rise to a radical c l a s s . THEOREM A: Let p denote any subradical. Then

(1) The c l a s s p = {R | p(R) - R J is a r a d i c a l c l a s s ; (2) For any ring R: p(R)c p(R);

(3) The following t h r e e statements a r e equivalent: (3a) p is a radical;

(3b) p is idempotent;

(3c) For any ring R: p(R) = p(R),

Proof: Let p be a subradical. Then, by definition, p a s s i g n s to any ring R an ideal p(R) of R in such a way that the conditions (A'), (B') and ( C ) a r e satisfied. F i r s t l y we show that (1) holds by proving that the c l a s s p = {R I p(R) = R } satisfies the conditions (A), (B) and (C). Assuming that R is a ring belonging to p and that « : R - S is a ring epimorphism we have, since p satisfies condition (A'), p ( S ) = p(a(R)) 3 a(p(R)) = a(R) = S . Hence S belongs to the c l a s s p . This p r o v e s that p satisfies condition (A). Now let R be any r i n g and let {Ail i el } be the collection of all ideals of R belonging to p. Denoting the sum of the ideals A i ( i e l ) by A it follows from the fact that p satisfies (B') that Ai c p(A) for every element i of I. By the defini-tion of the ideal A this implies that A c p(A), whence p(A) = A, or A e p . Obviously A is the greatest ideal of R belonging to p, whence we have shown that the class p satisfies condition (B). F o r proving that the c l a s s p satisfies condition (C) it suffices to show that the ring R/A does not contain a nonzero ideal belonging to p . Therefore, let I be an ideal of R, containing A, such that I/A belongs to p. Using that p satisfies condition (B') we obtain that A = p(A) c j6(I). Hence we have a canonical epimorphism I/A - l/p(I). Since the class p is homomorphically closed, as we have shown j u s t above, l / A s p i m -plies that l/p(I) e p, or j5(l/p(I)) = l/p(I). On the other hand we have that j6(l/j6(I)) = 0, since p satisfies condition ( C ) . Thus we obtain that I/i5(I) = 0, o r equivalently p(I) = I. Hence I s p. By the definition of the ideal A of R this implies that I c A. Consequently l / A = 0. This completes the proof of p a r t (1) of the theorem.

P a r t (2) of the theorem is almost trivial. F o r if R is any ring, then p(R) is the g r e a t e s t ideal of R belonging to p. Hence p(p(R)) = p(R). Since p satisfies condition (B') this implies that p(R) c p(R). It r e m a i n s to prove p a r t (3) of the t h e o r e m . The equivalence of the statements (3a) and (3b) is a consequence of the definitions. For a subradical i s a r a d i c a l if and only if it s a t i s f i e s condition (Bj[). If the subradical p is idempotent, then we have for any ring R: p(p(R)) = p(R). Hence p(R) belongs to p , whence p(R) c p(R). The opposite inclusion holds as well, by p a r t (2) of the theorem. Hence p(R) = p(R). So (3b) implies (3c). Finally, if p(R) = p(R) for any ring R, then p

assigns to any ring R the p r a d i c a l p(R) of R. As we astablished b e -fore, p satisfies the conditions (A'), (B'), (B!^) and ( C ) in that c a s e . Hence it follows that p is a radical. This completes the proof of the theorem.

On account of theorem A there exists a one-to-one correspondence between radical c l a s s e s and radical functions. In the future we shall not distinguish between a radical function p and the corresponding radical class p. F u r t h e r m o r e , any subradical p uniquely d e t e r m i n e s a radical p which coincides with p if and only if p is a radical. A radical p is called h e r e d i t a r y if and only if p, considered as a c l a s s of rings, is hereditary, i.e. any ideal of a ring belonging to p belongs to p. An important result is that the radical p is h e r e d i t a r y if and only if p(A) = A n p(R) for any ideal A of any ring R [19, t h e o -r e m 48]. A sub-radical j5 will be called h e -r e d i t a -r y if and only if p(A) = A n j5(R) for any ideal A of any ring R. F r o m the next theorem follows that any hereditary subradical is a radical.

THEOREM B: Let p denote any subradical, and let p be the r a d i c a l determined by p. Then the following t h r e e statements a r e equivalent: (1) p is a hereditary subradical;

(2) p = p and p i s hereditary;

(3) If A is an ideal of R such that A c p(R), then p(A) = A.

Proof: Let p be a hereditary subradical and let A be an ideal of any ring R. Then p(A) = A P, p(R). Hence, if A c p(R), then p(A) = A. This proves the implication (1) ^ (3). If (3) holds, then we have for any ring R p(p(R)) = p(R) since p(R) c p(R). Hence it follows from theorem A that p is a r a d i c a l and p = p. F u r t h e r m o r e , still a s s u m -ing that (3) holds, let R be a r-ing belong-ing to p and let A be an ideal of R. Then p(R) = R and consequently A c p(R). By (3) this implies that p(A) = A, or equivalently A e p, showing that the radical p is hereditary. This proves the implication (3) =* (2). The implication ( 2 ) ^ ( 1 ) being trivial, herewith the theorem is proved.

We proceed by listing some r e s u l t s concerning socalled upper r a -dicals. Let p be any radical. A ring R i s called p - s e m i - s i m p l e if p(R) - 0. The c l a s s of all p - s e m i - s i m p l e r i n g s will be denoted by Sp. For any radical p the c l a s s Sp is h e r e d i t a r y [19, corollary 2, page 125]. F u r t h e r m o r e , putting M = Sp the c l a s s M satisfies the follow-ing two conditions [19, theorem 2]:

(E) Every nonzero ideal of a ring of M can be mapped h o m o m o r -phically onto some nonzero ring of M;

(F) If every nonzero ideal of a ring R can be mapped h o m o m o r -phically onto some nonzero ring of M, then R belongs to M . Conversely, if M is a c l a s s of rings satisfying condition (E), then the c l a s s p, consisting of all rings R having no nonzero homomorphic

images in M, i s a radical c l a s s , and, m o r e o v e r , M c Sp. Note that if the c l a s s M is hereditary, then it trivially satisfies condition (E). Hence, if M is a hereditary c l a s s then M determines in the just d e s -cribed way a radical, which is called the upper radical determined by M and which is denoted by UM. Moreover, M c SUM. The word 'upper' is justified by the following property: If p is any radical such that M c Sp, then p c UM [19, l e m m a 4].

At this stage we need some r e s u l t s obtained in [26]. We shall say that a radical p has the intersection property with respect to the class N of rings if the p - r a d i c a l of any ring R equals the intersection of all ideals I or R such that R / l belongs to N. It is easily verified that the rings in N a r e p - s e m i - s i m p l e if p has the intersection property with r e s p e c t to N. F u r t h e r m o r e , if A i s an ideal of the ring R having non-zero intersection with every nonnon-zero ideal of R, then A will be called an essential ideal of R and R an essential extension of A. Obviously R is an essential extension of itself. R has no proper essential exten-sions if and only if R has an identity [26, theorem 2]. A c l a s s M of rings will be called essentially closed if it contains any essential e x -tension of every ring belonging to M. For any c l a s s M of rings the c l a s s Mk, consisting of all essential extensions of rings belonging to M, will be called the essential cover of M. Clearly M is essentially

closed if and only if M = Mj^. Now we have [26, theorem 7]: THEOREM C: Let M be a hereditary c l a s s of s e m i - p r i m e r i n g s .

Then the following t h r e e statements a r e equivalent:

(1) UM, the upper radical determined by M, is hereditary; (2) UM = UM^;

C H A P T E R I

A GENERAL TYPE OF REGULARITY FOR RINGS

1. DEFINITIONS

Let t h e r e be assigned to each ring R a mapping Fj^ which maps R i n -to the set S(R, +) of all subgroups of the additive group (R,+) of R. The c l a s s F, consisting of all mappings F R , will be called a

regu-larity for rings if the following t h r e e conditions a r e satisfied:

(Cl) if O!: R - S i s a ring epimorphism and if r e R then Fs(o!r) = a F R ( r ) ;

(C2) if A is an ideal of R and a e A then FA(a) c FR(a); (C3) if r , s e R and s e FR(r) then FR(r + s) c FR(r).

If F i s a regularity, an element r of R will be called FR-regnlar if r e F R ( r ) . R will be called Fregiilar if each element of R is F R -r e g u l a -r . The c l a s s of all F - -r e g u l a -r -rings will be denoted by F. An ideal A of R will be called F R r e g u l a r if each element of A is F R -r e g u l a -r , and F - -r e g u l a -r if each element of A i s F A - -r e g u l a -r , i.e. if A e F . We have, as a consequence of (C2), that each F - r e g u l a r ideal of R is F R - r e g u l a r . That the converse may fail to be t r u e is shown by the next example.

EXAMPLE 1: F o r each ring R and for each element r of R define F R ( r ) = R 2 . Then the c l a s s F = { F R | R is a ring} is a r e g u l a r i t y . If r e R then r is F R - r e g u l a r if and only if r e R 2 , whence an ideal A of R is F p - r e g u l a r if and only if A c R 2 and R is F - r e g u l a r if and only if R 2 = R. Now let R be J:he ring Z of the rational i n t e g e r s . Since Z has a unity element we have z 2 = Z. Hence each ideal of Z is F2;-regular. So is the ideal A = 2Z. Since A, considered as a ring, is not idempotent, the ideal A is not F - r e g u l a r .

2. T ^ E RADICAL AND THE SUBRADICAL DETERMINED BY A REGULARITY

Throughout this section F d e n o t e s an a r b i t r a r y , but fixed regularity. We shall show that F determines a radical F and a_subradical F . In fact F is the r a d i c a l determined by the subradical F .

F o r any ring R define the subset N(R) as follows: N(R) = { r e R J < r > is F R - r e g u l a r } , *'

i.e. N(R) consists of all elements of R generating an F R - r e g u l a r ideal of R. Our first r e s u l t is

*) F o r any subset S of R the ideal of R generated by S will be denoted by <S>.

THEOREM 1: For any ring R, N(R) is an ideal of R. Moreover, N(R) is the g r e a t e s t F R - r e g u l a r ideal of R.

Proof: F r o m the definition of the subset N(R) of R follows that each element of N(R) i s F R - r e g u l a r and that each F R - r e g u l a r ideal of R is contained in N(R). So the theorem will be proved if we can show that N(R) is an ideal of R, for then N(R) is clearly the g r e a t e s t FR-r e g u l a FR-r ideal of R.

Let a, b e N ( R ) . Then < a > and <b> a r e F R - r e g u l a r ideals of R, by the definition of N(R). Suppose that r e < a - b > . Since < a - b> c < a > + < b > we may write r = u + v for s o m e elements u and v be-longing to < a > and < b > respectively. Then r - v e < a > , and since < a > i s an F R - r e g u l a r ideal it follows that r - ve FR(r - v).Denoting the canonical epimorphism R - S, with S = R / < v > , by a, Q!(r -v) = a r implies that Fs(a(r - v)) = F s ( a r ) . Using (Cl) we obtain that

° ' F R ( ^ - V) = a F R ( r ) . Consequently FR(r v) c FR(r) + < v > . This i n -clusion together with r - v e FR(r - v) yields that r - v e F R ( r ) + < v > . Hence we may write r = y + w, with y e FR(r) and w e < v > . Since v e < b > implies that < v > c < b > , we have w G<b>. By the F R - r e g u l a r i t y of the ideal < b > it follows that w is F R - r e g u l a r , and hence r - y e F R (r - y). Now y e FR(r) implies that FR(r - y) c FR(r), by (C3). Hence r - y e FR(r). Since FR(r) is an additive group and ys F R ( r ) it follows that r e FR(r). This shows that the ideal < a - b > is F R - r e g u l a r . By the definition of N(R) this implies that a - b e N(R). For any element r of R we have < r a > c < a > and < a r > c < a > . Hence, since the ideal < a > i s F R - r e g u l a r , the ideals < r a > and < a r > are F R - r e g u l a r as well. Consequently r a e N(R) and a r e N(R). T h i s proves that N(R) i s an ideal of R, and completes the proof.

By virtue of t h e o r e m 1 any ring R contains a greatest F R - r e g u l a r ideal, namely N(R). The main result of this section is

THEOREM 2: The function Y, assigning to any ring R i t s greatest F R - r e g u l a r ideal, is a subradical.

Proof: We need to show that the function F , defined by F(R) = N(R) for any ring R, satisfies the conditions (A'), (B') and ( C ) . To begin with condition (A'), let a : R - S' be a ring homomorphism. Define S = aR. Then we have to show that Q!F(R) c F(S). Let r e F(R). Then r e FR(r) since the ideal F(R) is F R - r e g u l a r . Using that F satisfies (Cl) we obtain that 0iFR(r) = F s ( a r ) . Hence a r e Fs(ar). F r o m this it follows that the ideal_aF(R) of S is F s - r e g u l a r . By theorern 1 this implies that Oi:F(R) c F(S), which was to be proved. Hence F satisfies condition (A').

Now let A be an ideal of R such that F(A) = A. Then A, considered a s a ring, is F - r e g u l a r . Hence, if ae A, then a e FA(a). By (C2) this implies that a e FR(a), showing that the ideal A is FR-regular,whence it follows that A c F(R) by theorem 1^ This proves that F satisfies condition (B'). Finally_we show that F satisfies condition_(C'). Let R be any ring such that F ( R / F ( R ) ) is a nonzero ideal of R / F ( R ) . Then,

by the definition of F, the ring S = R / F ( R ) contains a nonzero ideal A / F ( R ) , where A is an ideal of R containing F(R), which is F s -r e g u l a -r . Let a s A. Then T-raeS, whe-re i-r denotes the canonical epimorphism R S. Since the ideal A/F(R) is assumed to be F s r e g u -lar, we have rrae Fs(Tra). Using (Cl) it follows that rrac TrFR(a). Hence

na. = ffb for some element b of FR(a). Since a - b e Ker(7r) = F(R), and

F(R) is F R - r e g u l a r , a - b e F R ( a - b ) . However, F R ( a - b ) c FR(a), by (C3), since b e FR(a). So we obtain that a - b e FR(a). Consequently, a e FR(a), because FR(a) i s an additive group containing the element b. This shows that the ideal A of R is F R - r e g u l a r . By theorem 1 thus follows that A c F(R), whence A / F ( R ) = 0. This is a contradic-tion. Hence the theorem is proved.

F o r ease of reference we state

COROLLARY 1: Every ring R contains a greatest F R - r e g u l a r ideal, denoted by F(R) and called the F-subradical of R^

F(R) is also the smallest ideal of R such that R / F ( R ) has zero F -subradical. *)

By noting that a ring R is F - r e g u l a r if and only if F(R) = R we obtain as a direct consequence of theorem 2 and theorem A the following THEOREM 3: The c l a s s F, consisting of all F - r e g u l a r rings, i s a radical c l a s s .

COROLLARY 2: Every ring R contains a greatest F - r e g u l a r ideal, denoted by F(R) and called the F-radical of R.

F(R) is also the smallest ideal of R such that R / F ( R ) has zero F -radical.

REMARK 1: For the proof of theorem 1 we only made use of the fact that F satisfies the conditions (Cl) and (C3).

Condition (C2) can therefore be omitted without loosing theorem 1. F u r t h e r m o r e , it is possible to show that if F merely satisfies the conditions (Cl) and (C3), then the function N, assigning to any ring R its greatest F R - r e g u l a r ideal N(R), satisfies the conditions (A') and ( C ) - This could have been expected on account of the work of BROWN and McCOY in [12] and [14]. Condition (C2) s e e m s to be a r a t h e r week condition however. AH examples given by BROWN and McCOY satisfy condition (C2). It must be noted that condition (C2) s e e m s to be n e c e s s a r y in proving that the c l a s s of all F - r e g u l a r rings is a radical c l a s s .

*) The second part of the corollary follows from the fact that F is a subradical. F o r if p is any subradical, then jo(R/A) = 0 implies that p(R) c A for any ideal A of any ring R. Taking for a the canon-ical epimorphism R - R / A this follows by applying condition (A').

3. COINCIDENCE OF THE RADICAL AND THE SUBRADICAL In this section the r e s u l t s concerning subradicals obtained in chapter 0 a r e applied to the subradical F determined by any regularity F . F will denote the radical determined by F again. As an immediate consequence of theorem A(2) we may state

I THEOREM 4: For any ring R: F(R) c F(R).

The inclusion F(R) c F(R) may be p r o p e r , as the following example shows.

EXAMPLE 2:_Let F be the regularity defined in example 1. Then we clearly have F(R) = R 2 for any ring R. Hence, the class F of all F -r e g u l a -r -rings consists of all idempotent -r i n g s .

Consequently, for any ring R F(R) is the g r e a t e s t idempotent ideal of R. Applying_these r e s u l t s to the ring R = 2Z of all even integers, we obtain that F(2Z) = 4Z and, since the ring 2Z does not contain nonzero idempotent ideals, F(2Z) = 0.

One may ask for conditions on the regularity F under which the F -radical and the F - s u b r a d i c a l coincide in every ring R. The next two t h e o r e m s a r e concerned with this problem.

THEOREM 5: Let F be any regularity. Then the following three statements_are equivalent:

(1) F(R) = F(R) for any ring R; (2) F(F(R))= F(R) for_any ring R;

(3) F(R) = 0 impHes F(R) = 0 for any ring R.

Proof: The equivalence of the statements (1) and (2) follows from theorem A(3). The implication (1) ^ ( 3 ) i s obvious. So it suffices to prove the implication (3) =>(1). Let R be any ring.Then F ( R / F ( R ) ) = 0 , by corollary 2. If (3) holds i^follows from this that F(R/F(R)) = 0. Now corollary 1 yields that F(R) c F(R).

Together with theorem 4 this implies that F(R) = F(R). This proves the t h e o r e m .

THEOREM 6: Let F be any regularity. Then the following three statements a r e equivalent:

(1) The subradical F is hereditary; _ (2) The radical F is hereditary and coincides with F ;

(3) If A is an ideal of R such that A c F(R), then F(A) = A. Proof: The theorem follows from theorem B.

Considering the equivalence of the statements (1) and (2) in theorem 6 we see that the h e r e d i t a r i n e s s of the subradical F implies the h e r e d i t a r i n e s s of the radical F. Below we give an example of a regularity F such that the radical F is hereditary and the subradical F is not. Because of the_equivalence of (1) and (2) we must have a proper inclusion F(R) c F(R) for at least one ring R for this r e g u l a r

-ity. This proves that the h e r e d i t a r i n e s s of the radical F does not imply the coincidence of the radical F and the subradical F . The fol-lowing question r e m a i n s :

PROBLEM 1: Does t h e r e exist a regularity F such that the F - r a d i c a l and the F - s u b r a d i c a l coincide in every ring R, while the radical F is not h e r e d i t a r y ?

We conclude this section by giving an example to show that the h e r e -d i t a r i n e s s of the r a -d i c a l F -does not imply the h e r e -d i t a r i n e s s of the subradical F.

EXAMPLE 3: Let F be the regularity determined by FR(r) = R r ( r + 1), Although it is not immediately clear that F is a regularity indeed we omit the proof of this fact; later on F will prove to be a so-called polynomial regularity. We show firstly that the radical F determined by F i s hereditary by proving that F(R) = 0 for any ring R. The proof is as follows. Let A be an F - r e g u l a r ideal of the ring R, and a e A. Since A is F - r e g u l a r we have a e Aa(a+ 1) = A(a+ 1) a c Aa. Hence a = ( -b)a, or equivalently (b+ l)a = 0, for some element b of A. The element b is F A - r e g u l a r too, so b e Ab(b + 1) c A(b + 1). This implies that b = c(b + l) for some element c of A. Now we have a = ( -b)a = = -c(b + l)a= -c. 0 = 0 . Hence A must be the zero ideal of R, so F(R) = 0. We proceed by constructing a ring S having nonzero F - s u b r a d i c a l . Let K[x] be the r i n g of polynomials in an i n d e r t e r m i n a t e x over an a r b i t r a r y field K, and let S be the quotient ring K[x] / < x 2 > , where <x2> i s the ideal of K[x] generated by x2. The elements of S can be r e p r e s e n t e d as polynomials in x of degree l e s s then two. Clearly Kx is an ideal of S. Since ox = ax(ax+ 1) for each element a of K, the ideal Kx is F s - r e g u l a r . Hence F(S) ^ 0.

This proves that the radical and the subradical determined by the regularity F do not coincide. Since F is hereditary we have as a consequence of t h e o r e m 6 that F is not h e r e d i t a r y .

4. EQUIVALENCE OF REGULARITIES

Two r e g u l a r i t i e s F and G will be called equal if FR(r) = GR(r) holds for any ring R and for any element r of R. Since different r e g u l a r -ities may determine the s a m e radical or the s a m e subradical we a r e led to the following

DEFINITION 1: Two r e g u l a r i t i e s F and G will be called strongly

equivalent if r e FR(r) holds if and only if r e GR(r), for any ring R

and for any element r of R, equivalent if F and G determine the s a m e subradical and weakly equivalent if F and G determine the s a m e radical.

REMARK 2: Obviously equality of F and G implies strong equivalence, and strong equivalence implies equivalence and weak equivalence. Moreover equivalence implies weak equivalence.

The next t h r e e examples serve to establish that the notions of equality, strong equivalence, equivalence and weak equivalence are d i s -tinct.

EXAMPLE 4: Consider the r e g u l a r i t i e s F and G determined by FR(r) = R and GR(r) = < r > respectively, where < r > i s the ideal of R generated by r . Each element r of any ring R is F R - r e g u l a r as well as GR-regular. Hence F and G are strongly equivalent, but

F and G a r e not equal.

EXAMPLE 5: Let F be the regularity defined in example 3. We proved that F(R) = 0 for each ring R, and F(S) f 0 for s o m e ring S. Hence F i s weakly equivalent but not equivalent to the regularity G determined by GR(r) = 0, for any ring R and for any element r of R. Before giving the next example we state

THEOREM 7: Let F be a regularity. Define GR(r) = F R ( - r ) for any ring R and for any element r of R. Then G = { G R j R is a ring} is a regularity. Moreover F and G a r e equivalent.

Proof: F i r s t we prove that G is a r e g u l a r i t y , i.e. satisfies the con-ditions (Cl), (C2) and (C3), by using that F does so. Let a : R - S be a ring epimorphism and let r e R. Then Gs(o;r) = F s ( - a r ) = Fs(a(-r)) = = a F R ( - r ) = 0!GR(r). If A is an ideal of R, then ae A implies GA(a) = = F ^ ( - a ) c FR(-a) = GR(a). Finally, if s e GR(r) for s o m e elements r and s of R, then - s e GR(r) since GR(r) is an additive group. Hence - s e F R ( - r ) , and therefore F R ( - r - s ) c F R ( - r ) , or G R ( r + s ) c GR(r), It r e m a i n s to show that F and G a r e equivalent.

Let R be a r b i t r a r y and r e F(R). Since F(R) i s an ideal we also have - r e F ( R ) . Hence - r e F R ( - r ) , because the ideal F(R) is F R - r e g u l a r . By the definition of G this implies - r e GR(r), and since_GR(r) is an additive group it follows that r e GR(r). Hence the ideal F(R) is GR-r e g u l a GR-r . Using coGR-rollaGR-ry 1 we conclude that F(R) c G(R). The pGR-roof of the inverse_inclusion i s s i m i l a r .

Thus we find F(R) = G(R), which proves that the r e g u l a r i t i e s F and G a r e equivalent.

EXAMPLE 6: Consider the r e g u l a r i t i e s F and G determined by FR(r) = R(l + r ) and GR(r) = R(l - r ) respectively. Then GR(r) = F R ( - r ) for any ring R and for any element r of R. Hence F and G are equi-valent by theorem 7. Now let K be a field with characteristic not equal to two. Denoting the unity element of K by e we have FK(e) = = K(l +e) = 2K = K and GK(e) = K(l - e) = 0. This shows that F and G a r e not strongly equivalent, because e e FK(e) and e e

GK;(e)-5. A SURVEY OF WELL-KNOWN REGULARITIES

In the ring theoretical l i t e r a t u r e many so-called r e g u l a r i t i e s appear. In this section we shall list those r e g u l a r i t i e s which a r e known to u s . One may easily verify that each of them satisfies the conditions (Cl), 18

(C2) and (C3), i.e. each of these wellknown r e g u l a r i t i e s is a r e g u l -arity in the general sense of section 1. We shall not c a r r y out this verification however; in the next chapter we shall introduce a c l a s s of regularities of a p a r t i c u l a r type, the socalled polynomial r e g u l -a r i t i e s . It will -appe-ar th-at -all well-known r e g u l -a r i t i e s -a r e poly-nomial r e g u l a r i t i e s .

1. FR(r) = Rr.

F - r e g u l a r rings were studied by BAER [8], by DIVINSKY [18] and by the author [41]. Following DIVINSKY this regularity will be called D-regularity, m o r e precisely left D-regularity.

2. FR(r) = r R r .

F r e g u l a r i t y in this sense is the well known von Neumann r e g u l -arity, first introduced by VON NEUMANN in [36]. See also [15]. 3. FR(r) = RrR.

In this case the c l a s s of all F - r e g u l a r rings coincides with the c l a s s X, introduced by DE LA ROSA in [43] and studied later on by the author [41].

4. FR(r) = R(l + r ) .

An element r of R is F R - r e g u l a r in this sense if r + s + s r = 0 for some element s of R. Hence F - r e g u l a r i t y coincides with left quasi-regularity, as defined by PERLIS [38] and later studied by BAER [8] and JACOBSON [27].

5. F R ( r ) = R r ( l + r ) .

In this case F - r e g u l a r i t y coincides with left pseudo-regularity as defined by DIVINSKY [17]. An element r of R is F R - r e g u l a r if r + s r + s r 2 = 0 for some element s of R.

6. FR(r) = Rr2.

F - r e g u l a r rings in this sense a r e the so-called strongly r e g u l a r rings as introduced by ARENS and KAPLANSKY in [7] and later studied by KANDO [29] and o t h e r s . See [32], [45] and [50]. 7. FR(r) = R r R r R .

Now F - r e g u l a r i t y coincides with f-regularity as defined by BLAIR [11], and later studied by ANDRUNAKIEVITCH [4].

8. FR(r) = R r + R r R .

The F - r e g u l a r rings a r e the rings whose homomorphic images have no nonzero right annihilators, as is proved by SZASZ in [48]. See also [42].

9. F R ( r ) = R ( l + r ) + R(l + r)R.

In this case the c l a s s F of all F - r e g u l a r rings coincides with the Brown-McCoy radical c l a s s . See [12] and [13].

lO.FR(r) = Rr + r R + R r R .

Here the c l a s s F coincides with the c l a s s EQ, consisting of all rings whose homomorphic images have no nonzero twosided

anni-hilators, introduced by SZASZ [49].

Finally we mention the class of so-called (p,q)-regularities, intro-duced by MCKNIGHT and also studied by GOULDING, MUSSER and ORTIZ in [25], [34], [35] and [37]. We postpone the description of these regularities to the next chapter. At this time we merely state that the (p,q) regularities are regularities in our general sense.

C H A P T E R I I

POLYNOMIAL REGULARITIES

1. INTRODUCTION

In chapter I we introduced a general type of regularity for rings. In the p r e s e n t chapter we proceed by giving a method to produce regu-l a r i t i e s . The r e g u regu-l a r i t i e s which a r i s e by using this method a r e the socalled polynomial regularities. Each of the wellknown r e g u l a r -ities listed in I. 5 will prove to be a polynomial regularity.

2. THE MAIN THEOREM; P-REGULARITIES

We s t a r t this section by stating the main theorem, namely

THEOREM 8: Let f i, f2, . . . , fn be a set of at least two polynomials over the rational i n t e g e r s . For each ring R define the mapping F R : R - S(R,+) a s follows:

FR(r) = fi(r)Rf2(r)R... Rfn.i(r)Rfn(r) *">

for each element r of R. Then F = { F R I R is a r i n g } is a regularity. REMARK 3: One may r a i s e a notational question. If r e R, where R is an a r b i t r a r y ring, and f = aQ + a^x + . . . + a^x^^ e Z[x], the ex-p r e s s i o n f(r) is not well defined since ag e Z and, ex-putting fo = ^-OLQ,

f o ( r ) e R . However, if s e R then sf(r) and f(r)s a r e well defined as ttQS + a ^ s r + . . , + ansr'^ and OLQS + a i r s + . . . a^r^^s respectively. T h e r e i s therefore no problem if f(r) is multiplied from the left or right by an element of R. If this is not the c a s e , as in the next l e m -ma for example, we shall consider f(r) as a for-mal expression which becomes meaningful only when multiplied by an element of R.

For the proof of t h e o r e m 8 we need t h r e e l e m m a s .

LEMMA 1: Let R be a ring, S a subring of R and r an element of R such that rS c S and Sr a S. If f e Z[x] and s e S, then f(r+ s) = f(r) + s ' for s o m e element s ' of S.

Proof: Let f be the polynomial aQ + a i x + . . . anx" (ai e Z, 0 ^ i ^ n). Then we have

*) As usual, for any two nonempty subsets A and B of R, AB denotes the subgroup of (R,+) generated by {abj a e A, b e B }. If A is a singleton {a}, then we write aB instead of {a}B.

So fi(r)Rf2(r)R.. . Rfn(r) denotes the subgroup of (R,+) generated by all elements of R of the form f i ( r ) r i f 2 ( r ) r 2 . . . rn_ifn(r), with I ' b ^2> •••> i - n - i e R .

f(r + s) = QQ + ai(r + s) + . . . cc^{r + s ) " = ag + a^r + . . . Oiy^r'^ + a j s + . . . = f(r) + a-^s + . . . .

The remaining t e r m s in the latter expression are multiples of finite products of r and s. Each of these t e r m s contains the element s as a factor at least once. Since s e S and S is a subring of R which is closed under left and right multiplication by r, it follows that each of these t e r m s belongs to S. Also a i s s S. Therefore f(r + s) = f(r) + s ' for some element s ' of S, which proves the lemma.

LEMMA 2: Let F be a regularity such that FR(r) i s an ideal of R for any ring R and for any element r of R. If moreover f,ge Z[x], then

GR(r) = f(r)FR(r)g(r) determines a regularity.

Proof: Let a : R - S be a ring epimorphism. Using the fact that F satisfies (Cl) we may write

aGR(r) = a(f(r)FR(r)g(r)) = f ( a r ) . a F R ( r ) . g(ar) = f(ar)Fs(ar)g(ar)= Gs(o!r),

proving that G = { G R | R is a r i n g } satisfies (Cl). If A is an ideal of R, then a e A implies

GA(a) = f(a)FA(a)g(a) c f(a)FR(a)g(a) = G R (a), since FA(a) e FR(a); hence G satisfies (C2).

Finally, to prove that G satisfies (C3), let s e GR(r).

Since FR(r) is an ideal of R we have GR(r) c FR(r), and therefore s -: FR(r). Since F satisfies (C3) this implies that F R ( r + s) c FR(r). Hence we have

GR(r + s) = f(r + s)FR(r + s)g(r + s) c f(r + s)FR(r)g(r + s). Clearly GR(r) is a subring of R. F u r t h e r m o r e , since r commutes with f(r) and g(r), we have rGR(r) c GR(r) and GR(r)r c; GR(r). Therefore, using l e m m a 1, s e GR(r) i m p l i e s that f(r + s) e f(r) + GR(r) and g(r+ s ) e g(r) + GR(r). Thus we find

f(r + s)FR(r)g(r + s) c (f(r) + GR(r))FR(r)(g(r) + GR(r)). The right m e m b e r of this inclusion is contained in

f(r)FR(r)g(r) + f(r)FR(r)GR(r) + GR(r)FR(r)g(r) + GR(r)FR(r)GR(r). By using the definition of GR(r) and the fact that FR(r) is an ideal of R one easily verifies that each of the t e r m s in the l a s t expression is contained in GR(r). Since GR(r) is additively closed t h i s proves that GR(r + s) c GR(r), and completes the proof.

REMARK 4: Lemma 2 can be strengthened. It suffices if F satisfies the following condition:

(C4) for any ring R and for any element r of R.

FR(r) is a subring of R such that r F R ( r ) c FR(r) and FR(r)r c FR(r).

In the following this strengthened form of lemma 2 will not be needed; we therefore omit the proof.

COROLLARY 3: Let f,ge Z[x]. Then FR(r) = f(r)Rg(r) d e t e r m i n e s a regularity. *)

Proof: Since ER(r) = R determines a regularity, and ER(r) is an ideal of R, for any ring R and for any element r of R, this i s a direct con-sequence of l e m m a 2.

LEMMA 3: Let F and G be r e g u l a r i t i e s such that FR(r) and GR(r) a r e ideals of R, for any ring R and for any element r of R. If f e Z[x] then

HR(r) = Fj^(r)f(r)GR(r) d e t e r m i n e s a regularity.

Proof: One may easily verify that H = { H R | R is a r i n g } satisfies (Cl) and (C2), by using the fact that F and G do so. We only prove that H satisfies (C3). Let s e H R ( r ) . Since FR(r) and GR(r) a r e ideals of R, we have HR(r) c FR(r) and HR(r) c GR(r).

Hence s e FR(r) and s e GR(r). Since F and G satisfy (C3), this i m -plies F R ( r + s) c FR(r) and GR(r+ s) c GR(r). It follows from this that

HR(r + s) = FR(r + s)f(r + s)GR(r + s) c FR(r)f(r + s)GR(r). By lemma 1 we have f(r+ s)e f(r) + HR(r), because HR(r) is an ideal of R and s e HR(r). Hence we may write

FR(r)f(r + s)GR(r) c FR(r)(f(r) + HR(r))GR(r).

Using the definition of HR(r) and the fact that HR(r) i s an ideal of R, it is easily verified that the right hand member of this inclusion i s contained in HR(r). Hence we have proved that HR(r +s) c HR(r). This shows that H satisfies (C3) and the lemma is proved. Now we a r e ready for the

Proof of theorem 8: By corollary 3 the theorem is t r u e for n = 2. We proceed by induction. Suppose that n > 2 and the t h e o r e m is t r u e for

*) The e a r l i e r mentioned (p,q)-regularities (see I. 5) a r i s e by using corroUary 3 and putting f = p and g = q.

each set of n - 1 polynomials, i.e. if g^, g2, . . . , gn_l e Z [ x ] , then GR(r) = gi(r)Rg2(r)R . . . R g n - l ( r ) determines a regularity. As a consequence of this assumption MR(r) = Rf2(r)R . . . Rfn_2(r)R d e -t e r m i n e s a regulari-ty, namely by -taking gj = g ^ . i = 1 and gi = f^ if 2 ^ 1 ^ n - 2 . As we have seen before ER(r) = R also d e t e r m i n e s a regularity. Since MR(r) and ER(r) are ideals of R, for each ring R and for each element r of R, we may apply lemma 3. This yields that NR(r) = MR(r)fn-l(r)ER(r) = Rf2(r)R . . . Rfn-l(r)R determines a regularity.

Since NR(r) is again an ideal of R, for any ring R and for any e l e -ment r of R, it follows from lemma 2 that FR(r) = fi(r)NR(r)fn(r) = = fl(r)Rf2(r)R . . . Rfn_i(r)Rfn(r) d e t e r m i n e s a regularity.

This proves the t h e o r e m .

DEFINITION 2: Each regularity which is of the type described in theorem 8, will be called a p-regularity.

REMARK 5: Returning to the list of well-known r e g u l a r i t i e s in I. 5, we see that seven of them are p - r e g u l a r i t i e s , namely left D - regu-larity, von Neumann reguregu-larity, X-reguregu-larity, quasi-reguregu-larity, pseudoregularity, strong regularity and fregularity. The r e m a i n -ing three seem to a r i s e by form-ing sums of p - r e g u l a r i t i e s in an ap-p r o ap-p r i a t e manner. For this reason the next section is devoted to the investigation of what we shall call the 'summability' of p - r e g u l a r i t i e s . 3. SUMMABILITY OF REGULARITIES; S P - AND P'-REGULARITIES DEFINITION 3: Let { F ( i ) I i e l } denote a family of r e g u l a r i t i e s . If

S R ( r ) = . i : ^ FR(i)(r) l e I

d e t e r m i n e s a regularity, this family will be called summable and S = { S R I R is a r i n g } will be called the sum of the r e g u l a r i t i e s F (^) ( i e l ) .

Not every family of regularities i s summable, as the following example makes clear. The crucial point is condition (C3). The class

S = { S R I R is a r i n g } , as defined above, satisfies (Cl) and (C2), but may fail to satisfy (C3).

EXAMPLE 7: Let F and G be the p - r e g u l a r i t i e s determined by FR(r) = Rr and GR(r) = r R respectively, and define HR(r) = Rr + r R for each ring R and for each element r of R. F u r t h e r m o r e , let S be the full ring of 2 X 2 m a t r i c e s over Z2, the p r i m e field of two e l e -ments, and let r be the element (f^ f^) of S. Then

Hence s = (Q Q) e Hs(r). Now r + s = (ri n) • Therefore t = (Q i) = l O / i o o ) belongs to S(r + s). Hence t e Hs(r + s). Since t / Hs(r) this shows that H s ( r + s ) jzf Hs(r), proving that H = { H R I R is a r i n g } does not satisfy (C3). So H is not a regularity. Consequently F and

G a r e not s u m m a b l e .

In the next t h e o r e m we shall give a sufficient condition for the s u m -mability of a family of p - r e g u l a r i t i e s . Let { F ( i ) | i e l } denote such a family. Then, for any element i of I, there exists a set of polynom-i a l s fpolynom-i(^), f2^-^^, . . . , fnpolynom-i^^^ (npolynom-i ^ 2) such that for any rpolynom-ing R and for any element r of R:

F R ^ ^ \ r ) = f / ^ \ r ) R f 2 ^ ' \ r ) R . . . R f n ^ ' \ r ) .

We shall say that the family { F ' ^ ^ J i s I } satisfies condition (S) if for any ring R and for any element r of R the following holds: if i,j e I and 1 < k < ni , then

f k ( ' \ r ) R F R ^ J \ r ) c S R ( r ) and FR^^\r)Rfk^^\r) c S R ( r ) , w h e r e SR(r) = E F p ^ ^ \ r ) .

" i e l " Now we can s t a t e

THEOREM 9: Any family of p - r e g u l a r i t i e s satisfying condition (S) is summable.

Proof: Using the notation introduced above we show firstly that SR(r) i s a subring of R for any ring R and for any element r of R.

Let i,j e l . Then

• F R ^ ' \ r ) F R ^ ^ \ r ) = f / ' \ r ) R . . . Rf^^^\(r)RfJ^\r)FR^J\r).

F^^' being a p - r e g u l a r i t y , we clearly have rFR^-''^(r) c FR^''^(r).

Consequently fhV(r)FR^^\r) c F R ^ ^ \ r ) . Hence it follows that F R ' ^ ( r ) F R ^ V ) c: f i ^ ' \ r ) R . . . Rf[^)_i(r)RFj^^J\r).

By applying (S) n i - 1 times on the right hand member of this inclusion we find

Therefore

FR'^(r)FR^V)cSR(r).

SR(r)SR(r)= Z F R ( ^ \ r ) F R ( i \ r ) c SR(r), proving that SR(r) i s a subring of R.

We proceed by showing that S satisfies (C3). As we have noted b e -fore this will prove that S is a regularity indeed. Let s e SR(r). Then we have to show that SR(r+ s) c SR(r). We s h a l l do this by showing that FR'^^(r + s) c SR(r) holds for every element i of L

We have

Fj^^^\r + s) = f / ^ \ r + s)R . . . RfjSj^r + s).

Since each F ^ "^ is a p - r e g u l a r i t y we have r F R (r) c F R (r) and F R (r)r c F R (r) for any element i of I.

Consequently rSR(r) c SR(r) and SR(r)r c SR(r). Hence s e SR(r) i m -p l i e s , by l e m m a 1,

f k ^ ' \ r + s) e fk^'^r) + SR(r), i e 1, 1 ^ k ^ ni. It therefore follows that

F R ^ ' V + s) c ( f / ^ \ r ) + SR(r))R . . . R ( f i J \ r ) + SR(r)). By reducing the right hand member of this inclusion we find

FR^^^r + s) c f / ^ \ r ) R . . . Rf^j^r) + . . . ,

FR^^^r + s) c

+ . . . ,

where the omitted t e r m s in the latter e x p r e s s i o n a r e products of the form X1X2 • . . X 2 n i - 1 , with X2k = R ( l ^ k ^ ni) and X2k-1 equal to SR(r) or to the polynomial fk(^)(r) (1 <; k ^ ni), whereas at least one of the odd factors is equal to SR(r). As a consequence of (S) we have that fk(i)(r)RSR(r) c SR(r) and SR(r)Rfk(i)(r) c SR(r) for every k (1 < k < ni) and also SR(r)RSR(r) c SR(r); the proof of each of these inclusions is straightforward and will t h e r e f o r e be left out. From this it follows that each of the omitted t e r m s i s contained in SR(r). Since FR(^)(r) is contained in SR(r) as well, it follows that FR(i)(r+ s) i s contained in SR(r).

This completes the proof.

REMARK 6: One may easily verify that condition (S) can be r e f o r m u l -ated as follows:

if i e I and 1 < k ^ n i , then

fk^'\r)RSR(r) c SR(r) and S R ( r ) R f k ( ' \ r ) c SR(r), for every ring R and for any element r of R,

by m e r e l y using the definition of SR(r). This makes clear that if fj^(i) = 1 for some i e I and for some k ( 1 ^ k <; ni), then condition (S) i s equivalent to the following condition:

(S') SR(r) is an ideal of R, for any ring R and for any element r of R. In general condition (S') is stronger than condition (S).

Thus we may state

COROLLARY 4: Any family of p - r e g u l a r i t i e s satisfying condition (S') is summable.

The next example shows that condition (S') is strictly stronger than condition (S) indeed.

EXAMPLE 8: Consider the four p - r e g u l a r i t i e s determined by r R r , r R ( r + l ) , '',r+l)Rr and (r + l ) R ( r + l ) respectively. This set of regu-l a r i t i e s satisfies (S), but does not satisfy (S').

THEOREM 10: Let { F ( i ) | i e l } denote a family of p - r e g u l a r i t i e s . For any ring R and for any element r of R define

S R ( r ) = . S F R ^ % ) . 1 € 1

Then

SR(r) = < S R ( r ) > ,

where < S R ( r ) > denotes the ideal of R generated by SR(r), determines a regularity.

Proof: The ideal < FR(^)(r)> of R, which is generated by FR(^)(r), equals

FR^^\r) + RFR^^\r) + Fj^^^\r)R + RFR^^\r)R.

Each of the four t e r m s in this sum determines a p - r e g u l a r i t y , since F(i) is a p - r e g u l a r i t y . Obviously SR(r) = . E^ < F R ( i ) ( r ) > . Thus we have ^ ^ '•

SR(r) = .L^ ( F R ^ ' \ r ) + R F R ^ ' \ r ) + F R ^ ' \ r ) R + R F R ( ' \ r ) R ) . l e I

Since SR(r) is an ideal of R, the theorem follows from corollary 4. Now we come to an easy, but important, application of theorem 10. In accordance with theorem 10 every p - r e g u l a r i t y F gives r i s e to another regularity, namely the regularity F ' d e t e r m i n e d by FR(r) = = < F R ( r ) > . The regularity F ' will be said to be generated by the p - r e g u l a r i t y F .

DEFINITION 4: A regularity which is the sum of a summable family of p - r e g u l a r i t i e s will be called an sp-regidarity. An s p - r e g u l a r i t y which is generated by a p - r e g u l a r i t y will be said to be d.

p'-regular-ity .

REMARK 7: The last t h r e e well-known r e g u l a r i t i e s listed in I. 5 a r e s p - r e g u l a r i t i e s . The r e g u l a r i t i e s 8 and 9 a r e p ' - r e g u l a r i t i e s . They a r e generated by the r e g u l a r i t i e s 1 and 4 respectively.

C H A P T E R I I I

OVERNILPOTENT AND UNDERIDEMPOTENT REGULARITIES

1. OVERNILPOTENT REGULARITIES

Let F be an a r b i t r a r y regularity, and let F denote the c l a s s of all F - r e g u l a r r i n g s . The regularity F will be called overnilpotent if any nilpotent ring is F - r e g u l a r , and supemilpotetit if, moreover, the c l a s s F is h e r e d i t a r y , i.e. if every ideal of an F r e g u l a r ring is F -r e g u l a -r .

PROPOSITION 1: The regularity F is overnilpotent if and only if /3 c F, where fi denotes the lower Baer radical.

Proof: This follows immediately from the fact that j3 is the smallest radical c l a s s containing all nilpotent r i n g s .

Since the radical p can also be c h a r a c t e r i z e d as the lower radical determined by the trivial ring ZQ on the additive group of the rational integers [19, page 43], we even have

PROPOSITION 2: The regularity F is overnilpotent if and only if F(Zo)= ZQ.

With the help of proposition 2 it is not difficult to determine whether or not a regularity is overnilpotent. This will be done in the next two sections for p - r e g u l a r i t i e s and for s p - r e g u l a r i t i e s respectively,

1. 1. OVERNILPOTENT P-REGULARITIES Let F be the p - r e g u l a r i t y determined by

FR(r) = fi(r)Rf2(r)R . . . Rfn(r),

where the fi (1 < i < n, n > 2) are a r b i t r a r y , but fixed, polynomials over Z. Then we can state

THEOREM 11: The following three statements concerning the p -regularity F a r e equivalent:

(1) F i s overnilpotent; (2) n = 2 and fi(0)f2(0) = + 1; (3) F is supernilpotent.

In case one of these conditions i s satisfied the subradical F is h e r e -ditary, and hence coincides with the radical F .

Proof: Suppose that (1) holds. By proposition 2 we then have F(Zo) = = Z Q . If n > 2, then FR(r) c R 2 for any ring R and any element r of R. Since Z Q i s F - r e g u l a r we obtain for any element r of Z Q : r e F z p ( r ) c z 2 = 0, which is a contradiction. Hence n = 2.

Now suppose that n = 2 and fl(0)f2(0) = m. Then, if r e ZQ, FzQ(r) = fl(r)Zof2(r) = fl(0)Zof2(0) = mZo- Therefore ZQ i s F - r e g u l a r if and only if ZQ = mZQ. Since mZQ = ZQ if and only if m = + 1, this proves the equivalence of (1) and (2). Next a s s u m e that (2) holds. Let A be any ideal of any ring R such that A c F(R), and_a e A. Then a = fl(a)rf2(a) for some element r of R, since a e F(R). Using the fact that fl(0)f2(0) = + 1, we obtain a = + r + . . . , where the remaining t e r m s a r e multiples of finite products of the elements a and r . Because a e A, and A i s an ideal of R, each of these t e r m s belongs to A. Hence a = + r + a' for some element a' of A, whence r e A. Therefore a e fl(a)Af2(a). This shows that the ideal A is F - r e g u l a r , i.e. A = F(A). By theorem 6 it follows that the subradical F is h e r e - , ditary and coincides with the radical F . Since (2) implies (1), the regularity F proves to be supernilpotent. Hence we have proved that (2) implies (3). The converse implication being trivial, this c o m -p l e t e s the -proof.

1.2. OVERNILPOTENT SP-REGULARITIES

Let F = . L^ F ' ^ ^ be an s p r e g u l a r i t y , where the F (^) (i e I) a r e p

-l e I , . v

r e g u l a r i t i e s . For each element i of I let F^-^' be determined by F R ^ ' \ r ) = f i ( ^ \ r ) R f 2 ^ ' \ r ) R . . . R f i | \ r ) , n^^ 2.

Let furthermore J denote the subset of I consisting of all elements i of I such that ni = 2 and fi(i)(0)f2(i)(0) ^ 0. Then we have

THEOREM 12: The s p - r e g u l a r i t y F i s overnilpotent if and only if J 7^<I>andgcd {fi(i)(0)f2(i)(0) | i e j } = 1.

Proof: Let F be overnilpotent and J = #. Then for each element i of I: n^ > 2 or f i(i)(0)f2(i)(0) = 0. Consequently FR(r) c R 2 for any ring R and any element r of R. Therefore, if R i s F - r e g u l a r , we obtain R c R 2 , or R = R 2 , contradicting the fact that any nilpotent r i n g i s F - r e g u l a r . Hence suppose that J 7^ $. Define d = gcd {fi(i)(0)f2ri)(0) j i e j } . Then, if r e Z Q , Zo2 = 0 implies that

Fz^oV) = 0 if i ^ J, and Fz[)'\r) = fi('\o)f2^'\o)Zo if ie J.

Hence we have

FZo(r) = . S ^ f / ' \ o ) f 2 ^ ' \ o ) Z o = dZQ. " l e J

Therefore ZQ is F - r e g u l a r if and only if ZQ = dZQ. Since ZQ = dZo if and only if d = 1, herewith the theorem i s proved.

Returning to the list of well-known r e g u l a r i t i e s in I. 5 we note that two of them a r e overnilpotent, namely q u a s i r e g u l a r i t y and the r e g u -larity introduced by BROWN and McCOY. T h e s e r e g u l a r i t i e s a r e

determined by FR(r) = R(l + r) and by FR(r) = R ( l + r ) + R(l + r)R respectively. The first regularity is a p - r e g u l a r i t y and it is clear from theorem 11 that it is supernilpotent. This fact is well-known and was first proved by PERLIS [39]. The second regularity is the p ' r e g u l a r i t y generated by the first, and by theorem 12 it is o v e r -nilpotent. It is a well-known fact however, that this regularity is actually supernilpotent. Both BROWN and McCOY [13] and SULINS-KY [47] proved that the radical determined by this regularity is h e r e d i t a r y . T h e i r proofs a r e r a t h e r involved and more or less artificial. We shall give a quite natural proof of a more general result. The key-point is the following

LEMMA 4: Let A be an ideal of R and a e A. If f e Z[x] and f(0) = 1 then a e Rf(a) + Rf(a)R if and only if A = Af(a) + Af(a)A.

Proof: The ' i f p a r t of the lemma i s trivial and it suffices to prove the 'only i f statement. Since Rf(a) + Rf(a)R is an ideal of R it fol-lows from a e Rf(a) + Rf(a)R that fo(a) c Rf(a) + Rf(a)R, where as usual fg = f - f(0). By using that A is an ideal of R we obtain that Afo(a)A c Af(a)A. Hence, if r , s e A, we have

r s = rf(a)s - rfo(a)s e Af(a)A + Afo(a)A c Af(a)A, proving that A 2 C AJ(a)A. Now suppose that b e A. Then it follows that

b = bf(a) - bfo(a) e Af(a) + A 2 c Af(a) + Af(a)A,

which proves that A c Af(a) + Af(a)A. Since the converse inclusion is obvious, this p r o v e s the lemma.

THEOREM 13: Let F be a supernilpotent p - r e g u l a r i t y . Then F ' , t h e p ' r e g u l a r i t y generated by F , i s supernilpotent. In fact the s u b -radical determined by F ' is hereditary.

Proof: Since the p - r e g u l a r i t y F is supernilpotent there must, by theorem 11, exist polynomials f and g in Z[x] such that FR(r) = f(r)Rg(r) for any ring R and any element r of R, and f(0)g(0) = + 1. Without loss of generality we may assume that f(0) = g(0) = 1. By the definition of F ' we have for any ring R and for any element r of R:

F ^ ( r ) = f(r)Rg(r) + Rf(r)Rg(r) + f(r)Rg(r)R + Rf(r)Rg(r)R. Theorem 12 makes clear that the regularity F ' is overnilpotent. Hence the theorem will be proved if we show that the subradical F ' determined by F ' i s hereditary. To do so, let A be an ideal of R such that A c F'(R) and a e A. It follows from a e F'(R) that ae F ^ ( a ) . Hence

a e f(a)Rg(a) + Rf(a)Rg(a) + f(a)Rg(a)R + Rf(a)Rg(a)R. F r o m this we deduce that

(1) a e R g ( a ) + Rg(a)R, and (2) aef(a)R + Rf(a)R.

By lemma 4, (1) implies that A = Ag(a) + Ag(a)A. In exactly the same way (2) implies that A - f(a)A + Af(a)A. Hence we may write:

A = f(a)A + Af(a)A = f(a){Ag(a) + Ag(a)A} + Af(a){Ag(a) + Ag(a)A} = f(a)Ag(a) + f(a)Ag(a)A + Af(a)Ag(a) + Af(a)Ag(a)A = F]^(a). Since a e A we obtain ae FR(a). Thus we have A = F'(A). By theorem 6 this shows that the subradical F ' is h e r e d i t a r y , which was to be proved.

COROLLARY 5: Let f e Z[x], f(0) = 1. Then the regularity F , d e t e r -mined by FR(r) = Rf(r) + Rf(r)R is supernilpotent. *)

2. UNDERIDEMPOTENT REGULARITIES

Let F be any regularity, and let F denote the c l a s s of all F - r e g u l a r r i n g s . The regularity F will be called underidempotent if every F -regular ring is idempotent, and subidempotent if, moreover, the c l a s s F is hereditary. The c l a s s Id, consisting of all idempotent r i n g s , coincides with the class of all G-regular rings, where G' is the p - r e g u l a r i t y determined by GR(r) = R 2 . Hence Id is a radical c l a s s . Clearly we have

PROPOSITION 3: The regularity F is underidempotent if and only if F c i d .

F u r t h e r m o r e , let Hid denote the class of all rings R having the property that each ideal of R is idempotent. Such rings a r e called hereditarily idempotent r i n g s . ANDRUNAKIEVITCH pointed out in [4] that the class Hid coincides with the c l a s s of all H-regular rings, where H is the well-known f-regularity introduced by BLAIR (see 1. 5). Thus Hid is a radical c l a s s .

The next proposition i s obvious.

PROPOSITION 4: The regularity F i s subidempotent if and only if F c Hid and F is hereditary.

We proceed to give n e c e s s a r y and sufficient conditions for a p - or s p - r e g u l a r i t y to be underidempotent, treating firstly p - r e g u l a r i t i e s and secondly s p - r e g u l a r i t i e s .

2. 1. UNDERIDEMPOTENT P-REGULARITIES Let F be the p - r e g u l a r i t y determined by

FR(r) = fi(r)Rf2(r)R . . . Rfn(r), *) This result was also obtained by ORTIZ [37].

where the fi (1 ^ i si n, n ^ 2) are a r b i t r a r y , but fixed, polynomials over Z. Then we can state:

THEOREM 14: The p - r e g u l a r i t y F is underidempotent if and only if either n > 2 or n = 2 and fi(0)f2(0) = 0.

Proof: If either n > 2 or n = 2 and fl(0)f2(0) = 0, then clearly FR(r) c R 2 for any ring R and any element r of R. Hence, if R is F - r e g u l a r , we have r e R 2 for every element r of R. Consequently R = R 2 , which proves that F is underidempotent. To prove the 'only i f part, a s -sume that n = 2 and fl(0)f2(0) = m f 0, and let Qo be the trivial ring on the additive group of the rational n u m b e r s . Then q c Q implies that FQo(q) = fl(q)Q0f2(q) = fl(0)Qof2(0) = mQo = QO- Thus Qo is F - r e g u l a r . Since QQ is not idempotent it follows that F i s not underidempotent in this case. This proves the theorem.

2. 2. UNDERIDEMPOTENT SP-REGULARITIES

Let F = .L., F^^' be an s p r e g u l a r i t y , where the F ' ^ ' (i cl) a r e p

-l e I

r e g u l a r i t i e s . Then we have

THEOREM 15: The s p - r e g u l a r i t y F is underidempotent if and only if F ( i ) i s underidempotent for each element i of I.

Proof: In case each F '^' is underidempotent we have for each element i of I that FR(i)(r) c R 2 for any ring R and for any element r of R, by theorem 14. Hence FR(r) = L FR(i)(r) c R 2 . From this it follows

i e I

that every F - r e g u l a r ring is idempotent, proving the ' i f part of the theorem.

If F ( i ) is not underidempotent for some element i of I, it follows from the proof of theorem 14 that the ring QQ is F(i)regular, which i m -plies that Qo is F - r e g u l a r . Hence F is not underidempotent in this c a s e . This p r o v e s the theorem.

COROLLARY 6: Let F be any underidempotent p-regularity. Then F ' , the p ' - r e g u l a r i t y generated b y F , is underidempotent.

C H A P T E R IV

UNIC POLYNOMIAL REGULARITIES

1. PRELIMINARY REMARKS

In chapter II we have been concerned with r e g u l a r i t i e s which can be described with the help of polynomials in Z[x], namely p - , p ' - and s p - r e g u l a r i t i e s . F r o m the p r e s e n t chapter on we shall r e s t r i c t our attention to such 'polynomial r e g u l a r i t i e s ' of a special type. Before starting our investigations we shall indicate which kind of r e g u l a r i t i e s shall be studied in the next. F i r s t l y we establish some notational con-ventions.

Let F be the p - r e g u l a r i t y determined by

FR(r) = f i ( r ) R f 2 ( r ) R . . . R f n ( r ) , where the fi (1 ^ i ^ n, n > 2) are polynomials over Z.

Then F will be denoted by {fiRf2R. . . Rfn}- Moreover, the subradical F(S) and the radical F(S) of any ring S wiU be denoted by (fiSf2S.. . Sfn) and [fiSf2S.. .Sfn] respectively. The p ' - r e g u l a r i t y F ' generated by F will be r e p r e s e n t e d by { < f i R f 2 R . . . Rfn>}j whereas the subradical F'(S) and the radical F'(S) of S wiU be denoted by ( < f i S f 2 S . . . Sfn> ) and [ < f l S f 2 S . . . Sfn> ] respectively.

The polynomial f over Z will be called unic if it has constant t e r m unity, i. e. f(0) = 1, and nullic if f = x^g for some natural number k and some unic polynomial g. As usual, f will be called a monomial if f = mx^ for some non-negative integer k and some integer m. A polynomial regularity for which all the arising polynomials over Z a r e monomials will be called a monomial regularity. For s h o r t n e s s sake we shall call monomial p , p ' and s p r e g u l a r i t i e s m, m ' -and s m - r e g u l a r i t i e s respectively.

The p - r e g u l a r i t y { f i R f 2 R . , . Rfn} ( p ' - r e g u l a r i t y { < f i R f 2 R . . . Rfn>}) will be called a imic p - r e g u l a r i t y (unic p ' - r e g u l a r i t y ) if each of the polynomials fi (1 ^ i -^ n) is unic, and a nullic p - r e g u l a r i t y (nullic p ' - r e g u l a r i t y ) if at least one of the polynomials fi i s nullic and the remaining polynomials a r e unic. An s p - r e g u l a r i t y will be called unic (nullic) if it i s the sum of unic (nullic) p - r e g u l a r i t i e s . Similar defini-tions hold for m - , m ' - and s m - r e g u l a r i t i e s .

If G, H and K a r e t h r e e r e g u l a r i t i e s such that for any ring R: K(R) = = G(R) n H(R), then the regularity K will be said to be the intersection of the r e g u l a r i t i e s G and H .

For any polynomial f over Z the p - r e g u l a r i t i e s {Rfl, {fR} and {RfR}, as well as the p ' - r e g u l a r i t i e s { < R f > } and { < f R > } will be called

The aim of the p r e s e n t chapter i s to show that each unic p - r e g u l a r i t y is the intersection of elementary (unic) p r e g u l a r i t i e s and a unic m regularity, and that each unic p ' r e g u l a r i t y is the intersecion of e l e -mentary (unic) p ' - r e g u l a r i t i e s and a unic m - r e g u l a r i t y . This will be done in the sections 2 and 3 r e s p e c t i v e l y . F u r t h e r m o r e we shall give s t r u c t u r e t h e o r e m s for the subradical determined by the elementary unic polynomial r e g u l a r i t i e s {Rf}, { < R f > } and {RfR} in the sections 4, 5 and 6 respectively.

(In the next chapter we shall consider nullic p - r e g u l a r i t i e s and nullic p ' - r e g u l a r i t i e s . It will be shown t h e r e that these r e g u l a r i t i e s a r e intersections of elementary r e g u l a r i t i e s and a nullic monomial r e g u l a r ity. Finally, chapter VI is devoted to the study of monomial r e g u l a r -ities).

2. AN INTERSECTION THEOREM FOR UNIC P-REGULARITIES We s t a r t with stating the main theorem of this section.

THEOREM 16: Let { f i R f 2 R . . . Rfn} be any unic p-regularity.Then we have for any ring R:

(flRf2R... Rfn) = (flR) n (Rf2R) n . . . n (Rfn-lR) n (Rfn) n R " ' ^

Moreover, (fiRf2R. • . Rfn) = R if and only if fi(r)Rf2(r)R. . . Rfn(r) = = R for any element r of R.

Note that the r e g u l a r i t i e s {flR}, {RfiR} ( 1 < i < n) and {Rfn} a r e elementary unic p - r e g u l a r i t i e s , whereas Rn-1 is the subradical of R determined by the unic m - r e g u l a r i t y { R ^ - l } . Hence theorem 16 makes clear that each unic p - r e g u l a r i t y is the intersection of some elementary unic p - r e g u l a r i t i e s and a unic m - r e g u l a r i t y . For the proof we need s e v e r a l l e m m a s .

LEMMA 5: Let f e Z [ x ] , f(0) = 1. Then we have for any ring R and any element r of R: r e Rf(r) if and only if R = Rf(r). *)

Proof: Obviously R = Rf(r) implies that r c Rf(r). If conversely r e Rf(r), let s be any element of R. Since the polynomial f is unic, we have s = sf(r) - sfo(r). Now fo(r) = t r for some suitable element t of R. Therefore sfo(r) = s t r e R r . F r o m this, together with r e Rf(r), it follows that sfo(r) e Rf(r). Since also sf(r) e Rf(r), we obtain that s e Rf(r). This p r o v e s that R c Rf(r). The converse inclusion i s ob-vious. Hence the l e m m a is proved.

Because of its importance in what follows we r e s t a t e l e m m a 4 in a weakened form.

LEMMA 6: Let f e Z[x], f ( 0 ) = l . Then we have for any ring R and any element r of R: r e Rf(r) + Rf(r)R if and only if R = Rf(r) + Rf(r)R.

• ) This lemma is due to MUSSER [35].