Ideals and radicals

IDEALS AND RADICALS

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IDEALS AND RADICALS 1 I * \

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IDEALS AND RADICALS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. R. VAN NAUTA LEMKE, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG

25 NOVEMBER 1970 TE 14.00 LTUR DOOR BENJAMIN DE LA ROSA MASTER OF SCIENCE GEBOREN TE SMITHFIELD

/^éo ^"/^^

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. F. LOONSTRA.

Hiermee betuig ek my dank aan die

SUID-AFRIKAANSE WETENSKAPLIKE EN NYWERHEIDNAVORSINGSRAAD

CONTENTS

CONVENTIONS 1

C H A P T E R I INTRODUCTION 3

C H A P T E R II PRIME IDEALS AND RADICALS 8

2.1. The radicals which contain the Baer-McCoy radical . . 8

2.2. A new radical class which contains p 9

C H A P T E R I I I QUASI-SEMI-PRIME IDEALS AND THE QUASI-RADICAL

OF AN IDEAL 13

3.1. Quasi-semi-prime ideals 13 3.2. The quasi-radical of an ideal 17

C H A P T E R IV QUASI-RADICAL RINGS AND THE BAER-MCCOY

RADICAL CLASS 19

4.1. The quasi-radical of a ring 19 4.2. Quasi-radical rings and the Baer-McCoy radical class. . 20

C H A P T E R V RINGS IN WHICH ALL IDEALS ARE QUASI-SEMI-PRIME 25

5.1. The A-radical of a ring 25 5.2. Quasi-semi-prime ideals and rings of matrices 31

C O N V E N T I O N S

By a ring we mean an associative ring, which, unless the contrary is stated, does not necessarily possess a unity and is not necessarily commu-tative. A two-sided ideal in a ring is referred to simply as an ideal. Rings and ideals are denoted by capital letters. A ring R with more than one element is called a simple ring if its only ideals are the trivial ideals (0)

andR.

The union of a set {Ai\ iel} of ideals in a ring is the set of all sums of elements from different Ai, each sum containing only a finite number of non-zero terms. The product of two ideals A and 5 in a ring is the set AB={Y,aibi\aieA,bieB}, it being understood that all finite sums of one or more terms are to be included. Products of more than two factors are defined inductively.

A ring R is said to be nilpotent if there exists a positive integer n such that R" = {Q)). If R^ = {0), then R is called a trivial ring. An element x of a ring is called a nilpotent element if there exists a positive integer n such that jc" = 0. A nil ring is a ring in which all elements are nilpotent.

C H A P T E R I

I N T R O D U C T I O N

The origins of ideal theory were closely linked up with the number theoretical notion of division, and accordingly the theory could not escape the influence of the fundamental concept of prime number. The desired counterpart was introduced by EMMY NOETHER in [15], where she defined the concept of a prime ideal in a commutative ring. Her two equivalent definitions may be formulated as follows:

DEFINITION 1.1. An ideal P in a commutative ring R is called a prime ideal if from abeP, where a, beR, it follows that aeP or beP.

DEFINITION 1.2. An ideal P in a commutative ring R is said to be prime, if the following condition is satisfied: If A and B are ideals in R such that AB^P, then Ac p or Be p.

The latter was taken over by KRULL [8 ] for the case of an arbitrary

ring, and this adoption has come to play an important role in the de-velopment of ideal theory.

Two further aspects which are important for our purposes are con-tained in KRULL [7]. Firstly, he observed that an ideal ƒ" in a commutative ring R is prime, if and only if the complement C(P) of P in /? is a multi-plicative system, that is, aeC{P) and beC{P) imply that abeC{P).

Secondly, he introduced the concept of a semi-prime ideal in a commu-tative ring: An ideal 5 in a commucommu-tative ring is called a semi-prime ideal if a"eS for some positive integer n, implies that asS. This definition, in which we may obviously take n = 2, leads in a natural way to the following generalization to the case of an arbitrary ring (cf. [13]).

DEFINITION 1.3. An ideal S in a ring R is called a semi-prime ideal if from A^^S, where A is an ideal of R, it follows that A^S.

MCCOY [11 ] proved that an ideal P of a ring R is prime if and only if the following condition is satisfied: If a and b are elements of R such that aRb^P, then aeP or beP. This fact inspired his generalization of

the concept of a multiplicative system to that of an m-system which he defined as a system M of elements of R with the property that aeM and beM imply that axbeM for some xeR. The importance of this concept lies in the fact that an ideal P in a ring R is prime if and only if the complement of P in i? is an m-system; an exact parallel of the role of multiplicative systems with respect to prime ideals in commutative rings. A further generalization in this respect was suggested by BROWN and MCCOY in [2] and carried out by the latter in [13] where he introduced the concept of an n-system to fill the same complementary role with respect to semi-prime ideals: An n-system N of a ring /? is a subset of R with the property that aeN implies the existence of an element x in R such that axaeN. McCoy proved that for every aeN there exists an 7M-system M in i? such that aeM and M^N. This relationship between wi-systems and «-systems represents a crucial point in his proof of the following important result on the relationship between semi-prime ideals and prime ideals.

LEMMA 1.4. An ideal S in a ring R is a semi-prime ideal if and only if S can be represented as an intersection of prime ideals of R.

In [11] McCoy defined the prime radical of an ideal A in a. ring R as the set of all elements r of R with the property that every w-system of R which contains r meets A. The set so defined for the zero ideal is called the prime radical of the ring R. He proved that this radical coincides with the intersection of all prime ideals in R. The prime radical coincides with the Baer lower radical introduced in [1]; a fact which was proved in-dependently by LEVITZKI [10] and NAGATA [14]. We shall refer to this radical as the Baer-McCoy radical and denote it by ^{R).

The theory of radicals has come a structure revealing way and since the coordinating definition by KUROSH in the early 1950's, this theory has matured into a confluence of elegance and generality. The definition referred to is included here for reference (cf. [4]).

Let (7 be a certain property that a ring may possess. A ring R is called a a-ring if it has the property a. An ideal ^4 of a given ring is called a a-ideal if A, viewed as a ring, is a a-ring. A ring which does not contain any non-zero c-ideals is said to be a-semi-simple.

DEFINITION 1.5. A property a is called a radical property if the following three conditions are satisfied:

(1) A homomorphic image of a a-ring is a a-ring.

(2) Every ring R contains a a-ideal S which contains every other a-ideal of R.

(3) The factor ring R/S is a-semi-simple.

The unique maximal «r-ideal S of i? is called the a-radical of R and is denoted by a(R). A a-ring is its own a-radical. Such a ring shall be termed a radical ring. The requirement (2) ensures that (0) is a a-radical ring with respect to any given a-radical property a. Obviously a ring is a-semi-simple if and only if a(R) = (0).

A given radical property and the class of all rings which are radical with respect to it, will be denoted by the same symbol. Radical properties are compared according to inclusion: If p and a are two radical properties, we shall write p < a to indicate the fact that the class of all p-radical rings is contained in the class of all a-radical rings. It can be verified that this condition is equivalent to either of the following two.

(i) p{R)^a(R) for every ring R.

(ii) Every a-semi-simple ring is p-semi-simple.

Two important types of radical properties have been constructed and widely used, the so-called lower and upper radical properties. The two constructions are briefly sketched below (cf. [4, 18]).

Let .5f be a non-empty homomorphically closed class of rings. Define JSfi=if. Assume that iP^, has been defined for every ordinal number a such that K a < ) ? , and define if^ to be the class of all rings R such that every non-zero homomorphic image of R contains a non-zero ideal which is a ring of if^, for some a.<fi. Let a = u ^ ^ , and call a ring a a-ring if it belongs to a. Then it can be verified that a is a radical property. Moreover, if a' is a radical property such that every ring in i f is a'-radical, then a^a'. The property a is called the lower radical property determined by the class S£.

Next let "^ be a class of rings with the property that every non-zero ideal of a ring of "^ can be mapped homomorphically onto some non-zero ring of %, and define W as the class of all rings R such that every non-zero ideal of R can be mapped homomorphically onto some non-non-zero ring of <%. Call a ring a p-ring if it cannot be mapped homomorphically onto any non-zero ring of 'U. Then p is a radical property, and if p' is

a radical property with respect to which all rings in '^ are semi-simple, then p'^p. Here p is called the upper radical property determined by the class <%.

In Chapter II we apply the concept of a prime ideal to the theory of radicals. We show that a given radical of every ring coincides with an intersection of prime ideals in the ring, if and only if the radical class concerned contains the Baer-McCoy radical class. This being the case with all the well known radicals in a ring, once again emphasizes the optimality of choice in Definition 1.2. As a further application we describe (in terms of prime ideals) the radical property p. which was recently introduced by JENKINS [6]. It is shown that p coincides with the upper radical property p* determined by the class of all simple non-trivial rings. Using this fact and results in [4, 9 ] we confirm the conjecture by Divinsky that the anti-simple radical class j5^ is properly contained in p*. At the same time we arrive at a more accurate positioning of p within the scheme of radicals. As an easy consequence of this positioning we conclude that p coincides with several well known radical properties in the class of commutative rings.

Returning once more to Definition 1.3 we note that it does not only represent a generalization of the concept of semi-prime ideal in a com-mutative ring to the same concept in an arbitrary ring, but also a gener-alization of the notion of a prime ideal in an arbitrary ring, for a prime ideal is semi-prime while the converse does not hold in general.

We offer yet another generalization of the concept of a prime ideal, as a matter of fact, our definition amounts to a weakening of the con-dition on an ideal to be semi-prime: An ideal Q of a ring R shall be called a quasi-semi-prime ideal if from RAR^Q, where A is an ideal in R, it follows that A^Q. If a is a fixed element of R we denote the set {xay\x,yeR} by RaR. It is shown in Section 3.1 that Q is quasi-semi-prime in R if and only if RaR^Q, where aeR, implies that aeQ. This criterion leads in a natural way to a generalization of the concept of an n-system to that of a t-system which we define as a system T of elements of R such that aeT implies the existence of elements x and y in R such that xaye T. It is an easy consequence of this definition and the mentioned criterion that an ideal Q in a ring R is quasi-semi-prime if and only if the complement of Ö in -^J is a /-system. We prove furthermore that the existence of a non-empty /-system of R which does not meet a given ideal in R, ensures the existence of at least one proper quasi-semi-prime ideal in R. Finally we show that ring homomorphisms associate quasi-6

semi-prime ideals with quasi-semi-prime ideals and that an ideal f in a ring R is quasi-semi-prime if and only if the zero ideal in the factor ring R/P is quasi-semi-prime.

Following the approach in [11] we define the quasi-radical q'{A) of an ideal A in a ring R as the set of all elements r of 7? with the property that every /-system of R which contains r meets A. Our main source of application in this respect is embodied in two characterizations of q'{A): We prove that q'{A) coincides with (i) the imique minimal quasi-semi-prime ideal of R which contains A, and (ii) the complement of the unique maximal /-system of R which does not meet A.

In Chapter IV we introduce the quasi-radical q(R) of a ring R, which we define as the quasi-radical of the zero ideal of R. It is an immediate consequence of our first characterization of q'{A), that q(R) coincides with the intersection of all quasi-semi-prime ideals in R. The major part of this chapter is devoted to a comparison between the quasi-radical property and the Baer-McCoy quasi-radical property. It is shown that the class of quasi-radical rings (i) is a hereditary class, (ii) contains the class of nilpotent rings properly, (iii) is properly contained in the Baer-McCoy radical class, and (iv) determines the Baer-Baer-McCoy radical property as a lower radical property.

Our final chapter is devoted to a study of the class of rings R with the property 1: 'all ideals of R are quasi-semi-prime'. We characterize such a ring R in terms of its elements by showing that the stated property is equivalent to the condition that every element a of R can be expressed as a finite sum of the form Y^^i^yi where X;, yieR. Using this charac-terization, we prove that A is a radical property. This radical property is shown to be non-hereditary and independent of each one in the sequence of best known radicals of a ring (cf. [4]). It is shown that the property is preserved under discrete direct sums and that the class of 1-radical rings contains several well known classes of rings, such as the rings with unity, the regular rings and the simple non-trivial ones. Several properties of rings with unity are shown to carry over to A-radical rings in general. Our main result in this respect is proved in the final section where we consider the ring R„ of nxn matrices over a ring R. We prove that all ideals of R„ have the form M„ where M is an ideal of R, if and only if /? is a i-radical ring.

C H A P T E R II

P R I M E I D E A L S A N D RADICALS

§2.1 The radicals which contain the Baer-McCoy radical.

The role of the concept of prime ideal in general ideal theory is well established. In the present chapter we consider this concept within the scope of a special class of ideals - the radical ideals. It is well known that the Baer-McCoy radical of a given ring coincides with the intersection of all the prime ideals in the ring (cf. [10, 11, 14]). The purpose of this section is to give an ideal-theoretic characterization of all radicals which contain this intersection. We shall need the following facts con-cerning an arbitrary ring R (cf. [13]).

(i) The Baer-McCoy-radical of R coincides with the intersection of all semi-prime ideals of R.

(ii) The intersection of an arbitrary set of semi-prime ideals of R is a semi-prime ideal of R.

THEOREM 2.1. Let a denote a radical property. Then the a-radical of every ring R is a semi-prime ideal of R, if and only if p^a.

Proof. Suppose that P^a. For an arbitrary ring R we have that a{R/a(R)) = {0), and hence p{R/a(R)) = (0). It therefore follows from (i) and (ii) that (0) is a semi-prime ideal in R/a(R). Let 6 be the natural homomorphism of R onto R/a(R) and let A be any ideal of jR such that

A^Cff{R) = K(e). Then A^e = (Aey=(0). Since (0) is a semi-prime ideal

in Rla{R) it follows that Ae = {0), and hence that Acff(R). Therefore a{R) is a semi-prime ideal in R. Conversely, if a{R) is a semi-prime ideal in R for every ring R, then it follows from (i) that P{R)^a{R) for

all rings R, that is, P^a. This completes the proof.

In view of Lemma 1.4 we have the following:

COROLLARY 2.2. The a-radical of every ring R coincides with an inter-section of prime ideals in R, if and only if P^a.

* 8

Turning to semi-simplicity we verify the following result regarding a given radical property a.

COROLLARY 2.3. Every a-semi-simple ring is isomorphic to a subdirect

sum of prime rings if and only if P^a.

Proof. Let a be a radical property such that every a-semi-simple ring is isomorphic to a subdirect sum of prime rings. It is well known that an ideal in a given ring is prime if and only if the zero ideal in the factor ring modulo that ideal, is a prime ideal. Hence it follows from the definition of a prime ring and Theorem 3.9 [13], that every a-semi-simple ring contains a set of prime ideals with zero intersection. Therefore all a-semi-simple rings are )S-semi-simple, so that P^a.

The sufficiency of the condition follows readily by applying Corollary 2.2 and arguments similar to those in the proof of the necessity.

§2.2 A new radical class which contains p.

JENKINS [6] has recently introduced a new radical property as follows: A proper ideal S o f a ring R is called a special ideal if R"^S for some positive integer n. If no such integer exists for S, then 5 is said to be non-special. A ring is called a p-ring if it has no non-special maximal ideals. Jenkins proved that a maximal ideal M is special if and only if R^sM, and that the property p is a radical property. The following lemma shows that this radical property may equally well be defined in terms of prime ideals.

LEMMA 2.4 A maximal ideal M of a ring R is non-special, if and only if it is a prime ideal of R.

Proof. If M is a prime ideal in R then /?^ $ M, as otherwise we would have that R = M, contradicting the maximality of M. Therefore M is non-special.

Conversely, suppose that M is a non-special ideal of J? and let A and B be ideals in R such that AB^M. Then A^M and B^M would imply that R^ = {A-i-M) (B-{-M) = AB+AM+ MB-\-M^^M, so that M would be special. Hence we have that A^M or BS:M, and accordingly Af is a prime ideal of R. This completes the proof.

Thus it follows that a ring is p-radical if and only if it has no prime maximal ideals.

Jenkins, has shown that p^p^v, where v denotes the Brown-McCoy radical property. In view of the preceding characterization, the first

inequality is immediate, for a ring is Baer-McCoy radical if and only if it has no proper prime ideals. On the other hand, if M is a modular maximal ideal of a ring R, (cf. [13]), then the simple ring R/M possesses a unity, and hence (R/MY = R/M. Therefore the zero ideal in R/M is prime, and it follows that M is a prime ideal in R. Thus it is clear that p^v.

The fact that p^p, implies that the /^-radical of an arbitrary ring R coincides with an intersection of prime ideals of R. To obtain a specified representation would be desirable, but we could not establish such a representation. At a first glance it seems as though p(R)^R may coincide with the intersection D of the prime maximal ideals of R. The inclusion p(R)^D is a consequence of the following lemma.

LEMMA 2.5. If a is a radical property such that P^a, then the a-radical of any ring R coincides with the intersection of all semi-prime ideals L j of R such that R/L^ is a-semi-simple.*

Proof. Theorem2.1 and the semi-simplicity of R/a{R)yield nL,• ^a{R). Conversely, let The any a-radical ideal in R. Then T+LJLiST/TnLi for each L;. Since T/TnL^ is a-radical, we must have that T+LJLi coincides with (0) in R/Li, so that T E L , . In particular a(R)^Li and hence a ( / ? ) £ n L j . Therefore a(R)=nLi. The proof is completed.

If Af is any prime maximal ideal in a ring R, then the zero ideal of the sim-ple ring i?/M is prime and maximal. Hence R/M is not i^-radical, and the simplicity imphes that p(R/M) = {0), that is, R/M is /i-semi-simple. Thus we have

COROLLARY 2.6. If R is a ring which is not p-radical, then p{R) is contained in the intersection of all prime maximal ideals in R.

To perceive the fact that the reverse inclusion does not hold in general, we consider the following example: Let S be a simple non-trivial ring without unity and let S' be the set of all pairs (/, a), where i is in the ring of integers and a in 5. Define addition and multiplication on S' as follows:

(/, a) 4- (j, b) = {i +j, a -\- b), (i, a) {J, b) = {ij, ib +ja + ab).

* This lemma, which replaces my direct proof of Corollary 2.6, is due to Dr. L. C. A.

van Leeuwen. I am indebted to him for this more general result.

Then 5 ' is a ring with unity (1,0), and the ideal {{0,a)\aeS} of S' is isomorphic to S. Identify S with this isomorphic ring. Then it can be shown that S is contained in every maximal ideal of S" (cf. [4]). Since S' has a unity, all its maximal ideals are prime. Hence S is contained in the intersection of the prime maximal ideals of S'. It readily follows that S does in fact coincide with this intersection. Thus, the assumption that p(S') coincides with the intersection of the prime maximal ideals of S', would yield the equality p{S') = S. This, however, would render S into a ^t-radical ring, contradicting its semi-simplicity with respect to p.

In order to obtain more information as to where p fits into the scheme of radicals, we characterize it as an upper radical property.

THEOREM 2.7. The property p is the upper radical property determined

by the class of all simple non-trivial rings.

Proof. Denote by £^ the class of all simple non-trivial rings and by p* the upper radical property determined by £/'. Then p* is the largest radical property with respect to which all rings in £/' are semi-simple. Since every ring in £^ is //-semi-simple, it follows that p^p*.

Conversely, let Rhe a ring which is not /t-radical. Then R contains a prime maximal ideal M and hence has the non-zero homomorphic image R/M in S^. Since 6^ consists solely of ^*-semi-simple rings, it follows that R is not p*-Tadica\. Thus we have that p*^p, and hence p=p*. This completes the proof of the theorem.

We are now in a position to confirm a conjecture by DIVINSKY concerning the anti-simple radical property p^ (cf. [4], p. 152). Recall that P^ is the upper radical property determined by the class of all subdirectly irre-ducible rings with idempotent hearts. Since all rings in 3^ are subdirectly irreducible and idempotent, (and coincide with their own hearts), it follows that P^^p*. However, results in [4] and [9] respectively, show that p^ is hereditary and p non-hereditary. By the preceding theorem we therefore have

COROLLARY 2.8. The anti-simple radical class P^ is properly contained in the radical class p*.

This corollary also yields an improvement on the positioning of p established in [6], for we know that p and the Levitzki radical class are both contained in p^. Furthermore, the independence of p and t,

the Jacobson radical property, established in [9], is also a consequence of our considerations. On the one hand, for instance, the zero ideal of a simple non-trivial Jacobson radical ring is prime and maximal, and on the other hand, we know that P^ and i are independent (cf. [4]). Confining ourselves to the class of commutative rings we may draw a further conclusion. For this class of rings we have that

P^ = i=v = S,

where ö denotes the upper radical property determined by the class of all matrix rings over division rings (cf. [4]). Since P^^p^v in general, we may therefore conclude as follows:

COROLLARY 2.9. For commutative rings, P^ = p = i=v = 5.

If C is a commutative ring which is not /i-radical, it follows by Corollary 2.6 and the relationship between prime and modular maximal ideals, that p{C) coincides with the intersection of all prime maximal ideals in C

in Theorem 3.3. Using this criterion, it is easily verified that the comple-ment of a proper quasi-semi-prime ideal in a ring i? is a /-system of R. In view of the fact that the trivial ideal R is quasi-semi-prime, we agree to consider the empty set 0 as a /-system, thus ensuring that the pre-ceding statement be true for all quasi-semi-prime ideals in R.

If conversely, Q is any ideal in R such that C(Q) is a /-system of R, it follows readily that Q is quasi-semi-prime in R. Thus we have the following characterization of a quasi-semi-prime ideal.

COROLLARY 3.6. An ideal Q in a ring R is quasi-semi-prime if and only if the complement C(Q) ofQ in R is a t-system of R.

Of course, there may exist many /-systems in a given ring R which do not coincide with the complements of ideals in R. Indeed, every ideal in R is a /-system of R. Moreover, there exist rings in which all non-empty /-systems contain 0. Rings possessing this property will be discussed in the next chapter.

Yet another frequently occurring phenomenon is the existence of a non-empty /-system in a ring R, which is contained in the complement of at least one ideal in R, but which does not necessarily coincide with such a complement. As the following lemma shows, the existence of a non-empty /-system in this general complementary setting, ensures the existence of at least one proper quasi-semi-prime ideal in R.

LEMMA 3.7. Let T be a non-empty t-system of a ring R and let A be an ideal of R which does not meet T. Then A is contained in an ideal M which is maximal in the set of all ideals in R which do not meet T. Such ideal M is necessarily quasi-semi-prime.

Proof. The existence of an ideal M with the required maximal property follows from an application of Zorn's Lemma to the set of all ideals in R which contain A but do not meet T. It remains to be shown that Af is a quasi-semi-prime ideal in R. Let a be an element of R such that RaR c M, and assume that a^M. Then the maximality of Af implies that the ideal M+{a) contains an element / of T; t=m + a', meM, a'e(a). Since T is a /-system of R, there exist elements x and y in R such that xtye T. It follows that xty=xmy + xa'y, and since xmyeM and xa'yeRaRzM, we have that xtyeM, and hence TnM^0. This contradiction shows that aeAf. Therefore M is quasi-semi-prime in R. This completes the proof. We conclude this section by describing the relationship between the

quasi-semi-prime ideals in a given ring R and those in an arbitrary homomorphic image of R.

THEOREM 3.8. If rj is a homomorphism of a ring R onto a ring S, with kernel K{ri), then:

(0 Jf Q '•' Ö quasi-semi-prime ideal in R which contains K{ri), then Qr\ is a quasi-semi-prime ideal in S.

(ii) If V is a quasi-semi-prime ideal in S, then Vri~^ = {reR\rrie V} is a quasi-semi-prime ideal in R, and K(ri)^ Vt]'^.

(iii) y]*'.Q-^Qr] is a one-to-one mapping of the set of all quasi-semi-prime ideals in R which contain K{ri) onto the set of all quasi-semi-prime ideals in S.

Proof, (i) Suppose that Q is a quasi-semi-prime ideal in R such that K{ri)^Q. Let ueS such that SuS^Qrj, and let aeR such that ari = u. If u^Qrj, that is, arj^Qrj, then a^Q. Since Ö is a quasi-semi-prime ideal in R, it follows that RaR^Q. Hence there are elements x and yinR such that xay^Q. We show that this exclusion is impossible. Since {xay)ri = {xrj) u{yt])eSuS, we have that {xay)r\eQi]. Consequently, {xay)r] = bt] for some element b in Q. Therefore {xay — b)t] = 0, and hence it follows that xayeb+K{r])S:b-\-Q = Q. Thus u does belong to Qr], and we have that Qri is a quasi-semi-prime ideal in S.

(ii) Suppose that F is a quasi-semi-prime ideal in S. Then Vt\~^ is an ideal in R, and K{r\)c, Vr]~^ (cf. [13]). Let A be any ideal in R such that RARcVr\-K Then {RAR)^'^{Vr\~^)^=V, that is, S{An)ScV. Since V is quasi-semi-prime in 5, it follows that Ar] £ V, and hence AS:Vr]~^. Therefore Vr]~^ is a quasi-semi-prime ideal in R.

(iii) By (i) we have that /j* is a mapping of the set of all quasi-semi-prime ideals in R which contain K{ri), into the set of all quasi-semi-quasi-semi-prime ideals in S. If, on the other hand, V is an arbitrary quasi-semi-prime ideal in S, it follows from (ii) and from the equality (yr]~^)r]=V, that V has a pre-image with the required properties. Hence r\* is an onto mapping. Since f/* is a restriction to quasi-semi-prime ideals of the one-to-one mapping I-^It] of the set of all ideals I'^K{ti) in R onto the set of all ideals in S, it follows that t]* is one-to-one. This completes the proof. In view of the fundamental theorem on ring homomorphisms, we may put S=R/K{ri). If K(ri) is a quasi-semi-prime ideal in R, it follows from (i) that (0) is quasi-semi-prime in R/K(r]). Conversely, if the zero ideal

in the factor ring R/K{r\) is quasi-Semi-prime, then condition (ii) implies that {0)ri~^ = K{ti) is a quasi-semi-prime ideal in R. Thus we have

THEOREM 3.9 An ideal Q in a ring R is quasi-semi-prime, if and only if the zero ideal in the factor ring R/Q is quasi-semi-prime.

§ 3.2. The quasi-radical of an ideal.

The present section is devoted to a discussion of a concept which closely resembles the notion of the prime radical of an ideal as introduced by MCCOY [11 ]. As a matter of fact, our definition amoimts to a restatement of that in [11 ], with /-systems taking the place of w-systems.

DEFINITION 3.10. The quasi-radical q'(A) of an ideal A in a ring R is the set of all elements r of R with the property that every t-system of R which contains r meets A.

It is clear from the definition that A is contained in its quasi-radical. We now characterize q'{AÏ) in two ways. Firstly, we describe it in terms of quasi-semi-prime ideals, and for the second characterization we shall once again employ the concept of a /-system.

THEOREM 3.11. The quasi-radical q' {A) of an ideal A in a ring R coincides with the intersection of all quasi-semi-prime ideals in R which contain A. Proof. We must show that q'{A) = r\Qi, where Qi runs through the set of all quasi-semi-prime ideals in R which contain A. Let 0 ^ {Oil-Then the assumption that q'{A)^Q yields the contradiction that at least one element of q'{A) belongs to the /-system C{Q), which does not meet A. Therefore q'{A)^Q, and we have that q'{A)^nQi.

Conversely, let reR such that r^q'{A). Then there exists a /-system T in R such that reT, and Tr\ A = 0. By Lemma 3.7 we have the existence of a quasi-semi-prime ideal Q' in R, with the properties A^Q' and Q'r\T=0. The first of these two properties shows that Q'e{Qi}. The second one and the fact that reT, ensure that r^Q'. Thus r^nQi, and we have that n Qi^q'{A). Therefore q'{A) — n Qj.

Using this theorem and Corollary 3.2, we obtain the following charac-terization of q'{A).

COROLLARY 3.12 The quasi-radical of an ideal A in a ring R is the unique minimal quasi-semi-prime ideal in R which contains A.

Before proceeding to our second characterization of q'{A), a further remark on /-systems is in order. The set C^ of /-systems of R which do not meet a given ideal A is non-empty, because 0eC^. Since the set-theoretic union of an arbitrary set of /-systems of /? is a /-system, it follows that the union of all elements of C^ is the unique maximal element of C^. We denote this maximal /-system by T{A).

THEOREM 3.13. The quasi-radical of an ideal A in a ring R coincides with the complement of the unique maximal t-system T(A).

Proof. The case where T{A) — 0 is trivial. Suppose that T{A)^0. Then there exists an ideal Min R with the following properties: (i) A^M, (ii) M is quasi-semi-prime in R, (iii) M is maximal with respect to MnT{A) = 0 (cf. Lemma 3.7). By (iii) we have that T{A)^C{M). Conditions (ii) and (i) show, however, that C{M) is a /-system of R which does not meet A, so that C{M)^T(A). Therefore C{M)==T{A), and hence M=C{T{A)). If A = M, then A is quasi-semi-prime, and we have q'{A) = A = C{T(A)), (cf. Corollary 3.12). Suppose that AczM, that is A(=C{T(A)), and assume the existence of a quasi-semi-prime ideal Q in R such that A^QcC{TiA)). Then C{Q) is a /-system of R, and T{A) czC{Q)^ C{A), contradicting the maximahty of T{A). Therefore A is not semi-prime, and C{T(A)) is the unique minimal quasi-semi-prime ideal in R which contains A. This completes the proof.

C H A P T E R IV

Q U A S I - R A D I C A L RINGS AND THE B A E R - M c C O Y RADICAL CLASS §4.1. The quasi-radical of a ring.

We continue our adoption of the approach in [11] in the following brief discussion.

DEFINITION 4.1. The quasi-radical of the zero ideal in a ring R may be called the quasi-radical of the ring R.

We denote this ideal of R by q{R) instead of by q'{{0)). By Definition 3.10, it is clear that q{R) consists of all the elements r of R which have the property that every /-system of R which contains r also contains 0. The following two characterizations are immediate consequences of Theorems 3.11 and 3.13 respectively.

COROLLARY 4.2. The quasi-radical of a ring R coincides with the inter-section of all quasi-semi-prime ideals in R. It is therefore the unique minimal quasi-semi-prime ideal in R.

COROLLARY 4.3. The quasi-radical of R coincides with the complement of the unique maximal t-system of R which does not contain 0.

The fact that {a, a^,...} is a /-system of R for every aeR, ensures that q{R) is a nil ideal in R. The following lemma yields a useful positioning of q{R) within the set of nil ideals of R.

LEMMA 4.4. The quasi-radical of a ring R is contained in the Baer-McCoy radical of R.

Proof. Since the Baer-McCoy radical of R coincides with the inter-Section of all prime ideals of R, and since every prime ideal in R is quasi-semi-prime, the required result follows at once from Corollary 4.2. To perceive the fact that these two ideals do not in general coincide, it is sufficient to consider the following elementary example.

EXAMPLE 4.5. Let R be the ring of integers modulo p", where p is a fixed prime, and n a fixed positive integer ^ 2 . Since R is commutative, P{R) consists of all the nilpotent elements of R, that is, the multiples of p modulo p". Clearly, however, q{R) = {0), since the zero ideal of R is

quasi-semi-prime.

As shall presently become evident, the quasi-radical of a ring is not a radical in the general sense of Definition 1.5. For this reason we shall not dwell upon its possibilities as such, but rather concentrate our attention on its relationship with general radicals. In the following section we discuss this relationship, with special reference to the Baer-McCoy radical property.

§ 4.2. Quasi-radical rings and the Baer-McCoy radical class.

Using the terminology of the general theory of radicals, we make the following definition.

DEFINITION 4.6. A ring R is called a quasi-radical ring if it coincides with its quasi-radical.

From Corollaries 4.2 and 4.3 respectively, we obtain the following two characterizations of a quasi-radical ring.

COROLLARY 4.7. A ring is quasi-radical if and only if it has no proper quasi-semi-prime ideals.

COROLLARY 4.8. A ring R is quasi-radical if and only if every non-empty t-system of R contains 0.

LEMMA 4.9. A nilpotent ring is quasi-radical.

Proof. Let J? be a nilpotent ring with R'' = {0). Then R''RR"^q{R), and since q{R) is a quasi-semi-prime ideal in R, it follows by a repeated application of Definition 3.1 that Rzq{R). Therefore R = q{R).

REMARK. P is the lower radical property determined by the class of all nilpotent rings (cf. [4]). Thus it follows by the preceding lemma and Lemma 4.4 that the quasi-radical property is not a radical property in the general sense of Kurosh. We now proceed to compare the quasi-radical property with general ones.

THEOREM 4.10. A homomorphic image of a radical ring is quasi-radical.

Proof. Let R he a quasi-radical ring and let S be an arbitrary homo-morphic image of R. By Corollary 4.7 we have that R is void of proper quasi-semi-prime ideals. Hence it follows from Theorem 3.8 (iii) that S is the only quasi-semi-prime ideal in S. Therefore S is quasi-radical. Next we derive a property of q{R) which is shared by all hereditary radicals.

LEMMA 4.11. If A is an ideal in the ring R, then the quasi-radical of the ring A contains the ideal A n q{R) of R.

Proof. If aeAnq{R), then aeq{R), and hence every /-system of R which contains a, also contains 0. Since all /-systems of A axe /-systems of R, it follows that every /-system of A which contains a, contains 0. Therefore aeq{A). This completes the proof.

On the other hand, however, the quasi-radical fails to satisfy the reverse inclusion, which holds for all general radicals. This can be seen by considering the nilpotent (and hence quasi-radical) ideal P{R) in Example 4.5. Here we have that q{p(R)) = p(R)ji=(0) = p(R)nq{R).

It is an immediate consequence of the lemma that q(R) is a quasi-radical ideal in R. The example shows, however, that q(R) does not always contain all quasi-radical ideals of R. Regarding semi-simplicity, a formal correspondence with general radicals remains: By Theorem 3.9 we have that the zero ideal in R/q(R) is quasi-semi-prime. Hence, using Corollary 4.2, we obtain q{R/q(R)) = {ö).

We shall now have a closer look at the class q of all quasi-radical rings. Firstly, we establish a basis for comparison between q and general radical classes.

LEMMA 4.12. Let a be a radical property. Then q is contained in the a-radical class if and only if q{R)^a{R) for every ring R.

Proof. The sufficiency of the condition is obvious. Suppose then that every quasi-radical ring is radical. Then every nilpotent ring is a-radical, (Lemma 4.9). Since P is the lower radical property determined by the class of all nilpotent rings, it follows that p^a, that is, p{R)Qa(R)

for every ring R. The required inclusion follows by Lemma 4.4. This completes the proof.

Although q is not a radical class we shall continue to write q^a to indicate the class inclusion stated in the lemma. From the proof of the lemma we may conclude as follows:

COROLLARY 4.13. q^aifandonlyifP^a; thus the complement of q in P is radically vacuous, so to speak.

Next we observe that if i? is a ring in q and I any ideal of R, then / is a ring in q. This is an immediate consequence of the fact that q{R) = R and the inchision Inq{R)S:q{I). Thus we have

COROLLARY 4.14. The class q is hereditary.

We already know that every nilpotent ring is quasi-radical. The following example taken from [13] shows that the converse does not hold in general.

EXAMPLE 4.15. Consider the rings S^ of integers modulo 2'"""^, {i= 1,2, 3,...), and denote by R^ the Baer-McCoy radical of 5j, that is, the multiples of 2 modulo 2'+i. Then R}+*=(0), whereas RiV(O) for k<i+l. Consider the discrete direct sum R of the rings i?;. Since each element of R differs from zero in only a finite number of components, every such element is nilpotent, and R is therefore a nil ring. However, for every positive integer n, there exist elements a in R such that a " # 0 . Therefore R is not nilpotent. We shall show that i? is a quasi-radical ring.

Let T be an arbitrary non-empty /-system of R and let ae T. If fl#0, then

a = (aj, «2,..., a„, 0,...), a^eR^, a„, # 0. Since T is a /-system of R, there exist elements

Xj = (rij, r^j,...), x'j = {r\j, r'jj,...), (j = 1, 2,...),

in R, such that x^ax'i e T, X2Xiax\x'2 e T, etc. Let k be the smallest positive integer such that 2k ^ m. Then

x„...X2Xiax\x'2...x'i, = (a[,a2,..., a'„,0, ...)eT, 22

where

a'i = '•,*••• ri2rnair'nr'i2 ... 4 , (j = 1, 2,..., m).

Since a,'ei?|''''=(0), it follows that OeT. Thus we have shown that every non-empty /-system of R contains 0, and hence that jR is a quasi-radical ring.

On the other hand of course, the class of quasi-radical rings is contained in the Baer-McCoy radical class (cf. Lemma 4.4). At this point we may mention that a ring satisfying the D.C.C. (or A.C.C.) on right ideals, is Baer-McCoy radical if and only if it is quasi-radical. This follows from the lemma referred to and from the fact that the Baer-McCoy radical of a ring with D.C.C. or A.C.C. on right ideals is a nilpotent ideal in the ring.

However, since the two classes do not coincide, a ring R with P(R) = R and q{R)cR must exist. The following example provides such a ring (cf. [4]).

EXAMPLE 4.16. Consider the set of all symbols x^, where ae(0, 1),

and let R he the commutative algebra over some field F with the x^ as a basis; multiphcation of these basal elements being defined by

x,,Xf = x^+p if a+P<l = 0 if oc-hP>i

It follows readily that i? is a nil ring, and being commutative, it is Baer-McCoy radical. However, R is not quasi-radical, as can be seen from the fact that the /-system

{^1/4 • ^1/4 = Xi/2, X^/g • Xi/2 ' ^ 1 / 8 '

^1/16 "^1/8 "^1/2 ' ^ 1 / 8 ' ^ l / i e » •••} >

for instance, does not contain 0, because

n 2" ~ ' 1

r = 2 Z

The preceding discussion yields the following refinement in the classi-fication of the nil rings in terms of hereditary classes.

THEOREM 4.17. The class of quasi-radical rings contains the class of nilpotent rings properly, and is properly contained in the Baer-McCoy radical class.

By Theorem 4.10 we have that the class q is homomorphically closed 23

and accordingly determines a lower radical property, a say. The first part of the preceding theorem assures us that every nilpotent ring is a-radical. Since P is the smallest radical property with respect to which every nilpotent ring is radical, it follows that jS<a. On the other hand, a is the smallest radical property with respect to which all rings in q are radical and hence, applying the second part of the theorem, we obtain a^p. Thus we have

THEOREM 4.18. The class of quasi-radical rings determines the Baer-McCoy radical property as a lower radical property.

As regards the actual construction of the lower radical class a in terms of q, (cf. Introduction), a further remark may be made. SHUKIN [17] has shown that the index a in the construction of an arbitrary lower radical property does not exceed «„, the first infinite ordinal number. In the case being considered, we observe that q contains all trivial rings. Moreover, q is hereditary. By Theorem 2, [18], we have therefore that the radical class a coincides with q2.

C H A P T E R V

R I N G S I N W H I C H A L L I D E A L S A R E Q U A S I - S E M I - P R I M E

§ 5.1. The l-radical of a ring.

The preceding chapter was to a considerable extent devoted to rings which are void of proper quasi-semi-prime ideals. In this chapter we shall exploit the other extreme and consider rings which contain nothing other than quasi-semi-prime ideals. These rings and the stated property which characterizes them, will be the topics of the present section.

The following characterization is of basic importance for our purposes.

THEOREM 5.1. For a ring R the following two statements are equivalent: (i) All ideals in R are quasi-semi-prime.

(ii) Each element a of R can be expressed as a finite sum of the form Y^XiCiyi, where Xi, yieR.

Proof. Suppose that every ideal in R is quasi-semi-prime and let a he an arbitrary element of R. Then the ideal RaR is quasi-semi-prime, and the trivial inclusion RaR^RaR implies that aeRaR.

Conversely, suppose that each element of R has property (ii) and let Q he any ideal in R. Then clearly RaR^Q implies that aeQ, and hence Q is quasi-semi-prime.

For convenience we introduce the following terminology: An element a of a ring R is called a X-element in R if a = Y,Xiayi for some x,, j,-6i?. The ring R is called a X-ring if every element of 7? is a A-element. An ideal y4 of a ring is said to be a X-ideal if ^4 is a A-ring. In view of Theorem 5.1 it is clear that the A-rings are exactly those in which all ideals are quasi-semi-prime.

The most obvious instance of a 1-ring is a ring with unity. Also, every regular ring R is a A-ring, because aeR implies that a=axa= {ax) a{xaxa) for some element x in R. Finally we note that every simple non-trivial ring 5 is a A-ring; indeed, SAS={0), where A is an ideal in 5, implies that A = (0).

the existence of a non-zero idempotent element in R, because 0#a:)c= (ax)^ for some element x in R. Thus the case of a simple non-trivial Jacobson-radical ring is an example of a A-ring which does not possess a unity and is not regular, for such a ring cannot contain any non-zero

idempotent elements. I We now consider the property A itself. We shall show that it is a radical

property and to this end we fit it into the scheme outlined in Definition 1.5. LEMMA 5.2. A homomorphic image of a X-ring is a X-ring.

The proof follows directly by applying the operation preserving properties of ring homomorphisms.

It is clear that the zero ideal of every ring is a A-ideal. To show that every ring contains a unique maximal A-ideal we verify the following result.

LEMMA 5.3. The union L of all the X-ideals of a ring R is a X-ideal in R. Proof. Let A and B he A-ideals in R and let s he an arbitrary element of A + B. Then s = a + b, where aeA and beB. Since ^ is a A-ideal in R, there exist elements X;, ƒ, in A such that a=^A-;a>'(. Denoting the element YjXi{a+b)yi by c, we can write:

a-\-b-c = a + b - ^ X i ( a -I- b) y^ = b -J^Xibyi. Hence it follows that a+b — ceB. Since 5 is a A-ideal in R, there exist elements Uj, Vj in B such that

a + b — c = Y,Uj{(i + b — c) Vj. It follows that

a + b = '£xiia + b)yi + '^Uj{a + b)vj I

Since clearly x,-, yi, Uj, Vj, UjXi, yiVjeA + B, we have that 5 is a A-element in A + B. Therefore A + B is a A-ideal in R.

Similarly it follows that the sum of any finite number of A-ideals of R is a A-ideal in R. Finally, since each element of the union of all A-ideals of R belongs to the sum of a finite number of these ideals, it is clear that every such element is a A-element in L. Therefore L is a A-ideal in R. This completes the proof of the lemma.

There remains to show that the factor ring R/L is A-semi-simple.

LEMMA 5.4. The factor ring R/L contains no non-zero X-ideals.

Proof. Let H/L he a A-ideal in R/L and let h + L be an arbitrary element of H/L. Then there are elements X;, ƒ,• in H such that

h + L= X(^.- + L) (/i + L) (ƒ,. + L) = Xxi%( + L.

This implies that h—Y^XihyieL, and since L is a A-ideal in R, it follows that h — Y,Xihyi=Y,Uj{h—Y,Xihy^ Vj for some Uj, VjeL. Therefore

h = 'Z Xihyi + E "jhVj - E Uj [X; Xjfeƒi] Dj-.

Since Uj, VjeL^H, it follows that Xi,yi,Uj,Vj,UjXi,yiVjeH. The last equality therefore shows that H is a A-ideal in R, and accordingly it must be contained in L. Therefore H—L, and H/L is the zero ideal in R/L. This completes the proof of the lemma.

By the preceding three lemmas we have

THEOREM 5.5. 77;^ property A is a radical property.

We denote the A-radical of a given ring R by X{K) instead of by L. Since X{R) is a A-ring, the element structure of this ideal is known by definition, and so is its ideal structure in view of Theorem 5.1. Regarding the relationship between X{R) and its annihilator

X{R)* = {aeR\a-X{R) = X{R)-a = (0)} in R, we prove the following:

LEMMA 5.6. A(^)nA(i?)* = (0), and X{R)* is a quasi-semi-prime ideal in R.

Proof. Let aeX{R)r\X{R)*. Since aeX(R) we have that fl=^Xjaj,- for some Xi,yfeX(R). By definition of X(R)*, however, it follows that (for instance) X;a = 0. Hence a = 0, proving the first part of the statement.

Let xeR such RxR^X{R)*, and let relX{R)* + {x)']nX{R), say r=l*+x', where l*eX{R)*, x'e(x). Since reX(R) we must have that r=^Uirvi for some MJ, VieX(R). Therefore

Hence reRxR^X{R)*, so that reX{R)nX{R)*={(i), and consequently [A(i?)* + (x)]nA(i?) = (0).

xeX{R)*, and we have that X{R)* is a quasi-semi-prime ideal in R. This completes the proof of the lemma.

It follows that q{R) is contained in X{R)* (cf. Corollary 4.2). In particular we have

COROLLARY 5.7. A quasi-radical ring is X-semi-simple.

We now examine the property A with respect to (i) the partition of the simple rings to which it corresponds, (ii) hereditariness, (iii) its relation-ship with other radical properties, and (iv) structure properties which it may yield.

From our examples we recall that the non-trivial simple rings are A-radical and hence they belong to the lower class with respect to A. The trivial simple rings, on the other hand, all belong to the upper class since they are quasi-radical, (Lemma 4.9), and hence A-semi-simple.

Since A is a radical property it satisfies the relation A(f)cƒ nA(-R) for an arbitrary ideal I in any ring R. The reverse inclusion, however, does not hold in general; for instance EnX{Z) = E^X{E) = {0), where E denotes the ring of even integers and Z the ring of all integers. Thus we have

COROLLARY 5.8. A is not hereditary.

REMARK. The above inclusion is a consequence of the following result by DIVINSKY and SULINSKY (cf. [4]): If a is any radical property, then for any ring R and any ideal I of R, a (I) is an ideal of R. The X-radical satisfies a stronger condition in this respect - any ideal I ofX{R) is indeed an ideal in R. In fact, if ael, then aeX[R), and hence a='^Xiayi for some Xi,yieX(R). For arbitrary reR we have that ar = Y,{Xia){yir). Since Xiael and yireX{R), it follows that (x^a) (j',r)6f, so that arel. Similarly rael.

Next we compare A with the well known radical properties / , where iS<Z<<^ = the upper radical property determined by the class of all fields (see diagram, p. 156, [4]). Since all fields are x-semi-simple and at the same time A-radical, it follows that X^x-On the other hand. Corollary 5.7 assures us that ^^A. Hence it follows by Corollary 4.13 that P^X,

and consequently x$A. Therefore A and x are independent radical properties.

This independence was to be expected, since the Baer-McCoy radical, for instance, is a measure for the presence of nilpotent ideals in a ring, while the A-radical measures the presence of 'well behaved' ideals, such as regular ideals and simple non-trivial ones. For this reason A may be considered as an anti-radical. Indeed, the A-radical of a ring R contains the Brown-McCoy anti-radical M(R), since M(R) is a regular ideal in R (cf. [3]).

Where semi-simplicity with respect to x is of special interest from a structural point of view, the emphasis must therefore be placed on radi-cality with respect to A. Our main result in this respect is proved in Section 5.2. Meanwhile we supplement Theorem 5.1 with some other properties of rings with unity which "carry over to A-radical rings in general. Firstly, we observe that every principal ideal (a) of a ring R coincides with the corresponding ideal RaR, if and only if i? is A-radical. This is an immediate consequence of the definition of A. Furthermore, a A-radical ring R is idempotent. In fact, the ideal R^ of R is quasi-semi-prime, and since R^^R^, it follows that RcR^, Therefore R^ = R. The following example shows however, that the A-radical rings do not exhaust the class of all idempotent rings: Let Af be the ring of 2 x 2 matrices over the ring of integers modulo 2, and let R he the subring of Af consisting of the elements

" 1 0 ] ^ r i 11

0

oj'

^

[o

oj"

Then R is idempotent, since, for instance, bR = R. However, R is not A-radical, for xay = 0 for all x, yeR. In fact, it is easily verified that R is A-semi-simple.

Finally we show that a A-radical ring R is isomorphic to the ring I of all its right multiplications. Recall that a right multiplication of R by an element a of J? is an endomorphism A, of the additive group R*, and that the image of an element xeR^ under a,, is the product xa in R. Moreover, the mapping n:a^a^ is a homomorphism of/? onto 1 (cf. [13]). Now aeK{n) implies that x a = 0 for all xeR. Since, however, a=Y_,Xiayi for some x,-, yieR, it follows that a=0. Hence n is an isomorphism. We conclude this section by providing the first of two methods for the construction of new A-radical rings from given ones. Lemma 5.3

-already suggests that the A-radical property is preserved under discrete direct sums. We give a direct proof of this fact.

THEOREM 5.9. The discrete direct sum of an arbitrary set of X-radical rings is a X-radical ring.

Proof. Let R he the discrete direct sum of A-radical rings R^, {iel), and let a he any element of R. Then a is a function defined on ƒ, such that the value a{i) belongs to Ri for each iel, and only a finite number of the a{i) differ from 0 (cf. [13]). Denote the non-zero components of a by a(/,.), ( 7 = 1 , 2 , . . . , / ) .

By the hypothesis on jRjj we have that

m

' j

fc=l

for some Xj^^, yi^^eRi^. Let m = ma\{mi^, Wj^,..., W;,). Then

m

k = l

where we takex,-^.t=j';jj = 0, (fc = m;j-|-l, Wi^ + 2 , . . . , m), for those indices ij where mi^Km. From (1) and the fact that a ( i ) = 0 for i^ij, we infer that

m

« = Z a. (2) t = i

where a,,{ij) = Xijk-a{ij)-yij^, and 0^(0 = 0 for / # ! ; , (/c=l, 2 , . . . , m). It follows that a^ = aia^^al, where al{ij) = Xi^k, at{ij) = a{ij), al{ij)=yiji„ and a'^(«) = a,fc(;) = a'^(j) = 0 for «V?j. Clearly, a4 = a and, using (2), we obtain

m

« = Z «*««*•

t = l

Therefore aeRaR, so that i? is a A-radical ring. This completes the proof of the theorem.

It must be observed that a regular ring already satisfies a stronger con-dition on ideals than the one required for a ring to be A-radical: Since an ideal P in a ring R is semi-prime if and only if aRa^P implies that aeP, it follows by the definition of a regular element that every ideal in a regular ring is semi-prime. A simple non-trivial ring satisfies an even

stronger condition, since both of the ideals in such a ring are prime. In view of the relation between prime, semi-prime and quasi-semi-prime ideals, both of these wider classes which contain the simple non-trivial rings and the regular rings respectively, are contained in the A-radical class. In order to see that these rings together with the rings with unity do not exhaust the A-radical class, we may consider the A-radical ring S=J®R, where / is a simple non-trivial Jacobson-radical ring and R is a ring with unity, containing a non-zero nilpotent ideal A. Since / does not possess a unity, S does not have one. Moreover, p{S) contains the nilpotent ideal {{0, a)\aeA}^{0), so that (0) is neither prime nor semi-prime in S (cf. Chapter II).

§ 5.2. Quasi-semi-prime ideals and rings of matrices.

In this section we apply the notion of a quasi-semi-prime ideal to rings

of nxn matrices over an arbitrary ring R. Such a ring of matrices will be denoted by the usual symbol R„. Although the ring R imder considera-tion need not possess a unity, we still use the matrix units E;J in a formal way: If xeR, then xEu is to be interpreted as the matrix in R„ with the element x at the intersection of the i'^ row andy"" column, and the zero element of R in the other positions (cf. [13]). The following lemma can be proved by direct calculation.

LEMMA 5.10. Letp, r, s andq be arbitrary integers from theset {1,2,...,n}, (x = Y,(iijEij any element of R„, and x and y arbitrary elements ofR. Then

(xEpr) ( Z ^iAj) ( y^sq) = xa,, yEp^.

We shall use this lemma to derive some results concerning the relationship between the quasi-semi-prime ideals in a given ring R and those in the corresponding ring R„.

THEOREM 5.11. An ideal Q in a ring R is quasi-semi-prime if and only if Q„ is a quasi-semi-prime ideal in R„.

Proof. Suppose that g is a quasi-semi-prime ideal in R and let a = Z dijEij he any element of i?„ such that R„aR„^Q„. lfafQ„, then a^^^Q for some k, W6{1, 2 , . . . , « } . Since Ö is a quasi-semi-prime ideal in R, we have that Ra^mR^Q, that is, there exist elements x and y in R such that xa^„y^Q. But if this was the case it would follow from Lemma 5.10 withp = r=k and s=q=m, that

(xEni) a ( yE„J = xa^„ yE^„ ^ Q„.

However, this is impossible, since R„aR„sQ„. Therefore ixeQ„, and we have that Q„ is quasi-semi-prime in R„.

Conversely, suppose that Q„ is a quasi-semi-prime ideal in R„, and let aeR such that RaRsQ. We shall show that R„yR„^Q„, where y=Y,aEij. An arbitrary element of R„yR„ is a finite sum of elements of the form

r' = ( Z ^ ü £ ü ) ( Z « £ ü ) ( Z > ' ü Ê ü ) . which is the sum of «* matrices of the form

{XprEpr) ( Z « ^ i j ) ( ysqE.q) = Xp^aysqEpq •

Since RaR^Q, it follows that Xp^ay^^eQ, and hence y'eQ„. Therefore R„yR„^Q„. Since Q„ is a quasi-semi-prime ideal in R„, it follows that y = YjaEijeQ„, and thus that a e ö - Therefore Q is a quasi-semi-prime ideal in R. This completes the proof of the theorem.

A ring R„ of matrices may of course contain ideals which are not of the form ƒ„ where / is an ideal in the ring JR. For instance, in the ring E2 of 2 x 2 matrices over the ring of even integers, the set

4k a

b c keZ; a, b, ceE

is such an ideal. We shall prove that this phenomenon cannot occur in a A-radical ring and, moreover, that the A-radical class contains exactly all rings R for which A-^A„, where A denotes an ideal in R, is a lattice isomorphism between the lattices of ideals in the rings R and R„ respec-tively. For this purpose we shall need the following fact (cf. [13]).

LEMMA 5.12. IfJf is an ideal in the ring R„ then the set M of all elements at the intersections of the first rows and first columns of matrices in Ji is an ideal in R.

THEOREM 5.13. The ideals of the ring R„ are of the form M„, where M is an ideal in R, if and only if R is a X-radical ring.*

Proof Suppose that Ris a A-radical ring. Let J( be an arbitrary ideaj in R„, and let Af be the ideal in R associated with J( as in Lemma 5.12_ * I have proved this theorem as a generalization of Theorem 2.24 in [13], unaware of the fact that part of the result was already known. Another proof of the sufficiency of the condition is contained in [5].

We show that Jt = M„. Let a = Yj^ijEijeJ(. Then, for arbitrary x, yeR, it follows from Lemma 5.10 mfhp = q=l that

xa^s yE, 1 = (x£i,) ( Z a^Êy) ( J'^si) e ^ •

Thus, by definition of Af, we have that xa^jeAf. Since this is true for arbitrary x,yeR, it follows that Ra^^R^M, and the fact that Af is a quasi-semi-prime ideal in R ensures that a^^eM. The latter relationship, being true for all r, se{l, 2 , . . . , «}, yields the fact that a.eM„. Therefore J^^M„.

If, on the other hand, m is any element of Af, then there exists a matrix Z ^ ^ O ' ^ Ü ^^ "^ ^'^'^ mii = m, and by Lemma 5.10 with r=s=l, it follows that

xm 11 yEp^ = (x£pi) ( ^ m;j-£ij.) ( j;£i,) e . ^ , that is,

xmyEp^ e J(,

where x and y are arbitrary elements of R. Thus every finite sum of the form ^XiW^i^p^, (Xi,jiejR), belongs to Ji. Since i? is A-radical, it follows that mEp^eM. Thus, for every meM, the ideal M contains the n^ matrices mEp^; p,qe\l,2, ...,n^. Consequently M„'^Jt, and we have that Ji = M„. This proves the sufficiency of the condition.

Conversely, suppose that every ideal in i?„ has the form M„, where M is an ideal in R, and let A he any ideal in R. Then the sets jSf and 01 of matrices in R„ with entries running through the ideals A, RA and AR as indicated in

~A AR...AR~ A AR...AR A AR...AR_

respectively, are obviously two-sided ideals in R„. By the hypothesis on R„ it follows that Se==Si = A„ = {RA)„ = {AR)„. Hence we have that A = RA = AR.

If Q is an arbitrary ideal in R and if A is any ideal in R such that RAR^Q, then it follows by the preceding equalities that A=AR= RAR^Q. Therefore Q is a quasi-semi-prime ideal in JR. Since Q is arbitrary, we have that R is a A-radical ring. Thus the necessity of the condition is proved.

By the preceding two theorems we obtain the following method for the construction of new A-radical rings from given ones:

COROLLARY 5.14. 77ie ring R„ is X-radical if and only if R is X-radical. A A ... A

RA RA...RA _and RA RA...RA

REFERENCES

BAER, R . , Radical Ideals, Amer. J. Math., 65 (1943), 537-568.

BROWN, B . and M C C O Y , N . H . , Prime ideals in nonassociative rings. Trans. Amer. Math. S o c , 89 (1958), 245-255. ^

BROWN, B . and M C C O Y , N . H . , The maximal regular ideal of a ring, Proc. Amer. Math. S o c , 1 (1950), 165-171.

DrviNSKY, N . J., Rings and Radicals, Univ. of Toronto Press, (1956).

JACOBSON, N . , Structure of rings, Amer. Math. S o c , CoUoq. Pub. vol. 37, Providence, 1956.

JENKINS, T . L . , A maximal ideal radical class, J. Nat. Sci. and Math., 7 (1967), 191-195.

KRULL, W . , Idealtheorie in Ringen ohne Endlichkeitsbedingung, Math. Ann., 101 (1929), 729-744.

KRULL, W . , Zur Theorie der zweiseitige Ideale in nichtkommutativen Bereichen, Math. Zeitschr., 28 (1928), 481-503.

LEAVITT, W . G . and JENKINS, T . L . , Non-hereditariness of the maximal ideal

radical class, J. Nat. Sci. and Math., 7 (1967), 202-205.

LEVITZKI, J., Prime ideals and the lower radical, Amer, J. Math., 73 (1951), 25-29.

McCoy, N . H., Prime ideals in general rings, Amer. J. Math., 71 (1949), 823-833. M C C O Y , N . H . , Rings and ideals, Carus Monograph Series, no. 8, Math. Assoc. of America, (1965).

M C C O Y , N . H . , The theory of rings, Macmillan, New York, (1964).

NAGATA, M . , On the theory of radicals in a ring, J. Math. Soc. Japan, 3 (1951), 330-344.

NOETHER, E . , Idealtheorie in Ringbereichen, Math. Ann., 83 (1921), 24-66. SASIADA, E . and COHN, P. M., An example of a simple radical ring, J. Algebra, 5 (1967), 313-i77.

SHUKIN, K . K . , On radical theory in groups, Sibirski Math. J. 3 (1962), 932-942. SuLiNSKi, A., ANDERSON, R . and DIVINSKY, N . J., Lower radical properties for

associative and alternative rings, J. London Math. S o c , 41 (1966), 417-424.

S A M E N V A T T I N G

Dit proefschrift heeft als uitgangspunt het concept van een priemideaal, en als toepassingsgebied de theorie van radicalen.

Met betrekking tot een gegeven radicaal-eigenschap a wordt onderzocht onder welke (noodzakelijke en voldoende) voorwaarden het a-radicaal van elke associatieve ring R kan worden voorgesteld als een doorsnede van priemidealen in R (zie Hoofdstuk II). Als voorbeeld van een radicaal met deze eigenschap bestuderen wij de (onlangs ingevoerde) radicaal-eigenschap p van Jenkins. Er wordt bewezen dat p samenvalt met de boven-radicaal-eigenschap p* die bepaald wordt door de klasse van niet-triviale enkelvoudige ringen. Deze karakterizering leidt tot de be-vestiging van het vermoeden van Divinsky dat p^<p*.

In Hoofdstuk III worden quasi-semi-priemidealen (als generalizatie van priemidealen) met hun eigenschappen behandeld. De klasse q van quasi-radicaal-ringen, (d.z. ringen zonder echte quasi-semi-priemidealen), wordt in Hoofdstuk IV nader onderzocht. Er wordt bewezen dat q een erfelijke klasse is die de klasse van nilpotente ringen echt omvat en de Baer-McCoy radicaal-eigenschap niet-triviaal bepaalt als onder-radicaal-eigenschap.

Het laatste hoofdstuk is gewijd aan de klasse A van ringen waarin alle idealen quasi-semi-priem zijn. Deze klasse wordt gekenmerkt als een niet-erfelijke radicaal-klasse. Het betreffende radicaal blijkt onafhankelijk te zijn van alle radicalen x met de eigenschap ) S < x ^ ^ - Als toepassing wordt de ring R„ van (/i x «)-matrices over een ring R beschouwd. Er wordt bewezen dat alle idealen van /?„ van de vorm M„ zijn, dan en slechts dan als R tot A behoort, (Af stelt een ideaal in R voor).

B I O G R A P H Y

The author of this thesis was born in South Africa in 1934. He matriculated from the Smithfield High School in 1953. After a period of alternative work and study, he obtained the degree of M.Sc. at the University of the Orange Free State in 1962. He was appointed lecturer at the same university in 1963 and promoted to senior lecturer in 1967. In 1969 he was appointed on the scientific staff of the University of Technology, Delft.

STELLINGEN

I

De bewering van N. J. DIVINSKY dat het Baer-McCoy radicaal bevat is in het boven-radicaal, bepaald door de klasse van niet-triviale enkel-voudige ringen, is juist. Het argument dat hij gebruikt om deze bewering waar te maken, is echter onaanvaardbaar.

N. J. DIVINSKY, Rings and Radicals, University of Toronto Press, 1965.

II

L. FucHS beweert ten onrechte dat de volledige directe som van torsie-vrije groepen van rang 1 en hetzelfde type een homogene groep is.

L. FucHS, Abelian Groups, Publishing House of the Hungarian Academy of Sciences, Budapest, 1958.

Ill

De resultaten van J. CHAUDHURY en W. N. EVERITT aangaande het punt-spectrum van een zelf-geadjungeerde gewone tweede-orde-differentiaal-operator, zijn met behulp van de door hen gebruikte funktie-theoretische methoden aanzienlijk korter af te leiden.

J. CHAUDHURY and W. N. EVERITT, On the

Spec-trum of Ordinary Second Order Differential Oper-ators, P.R.S.E., vol. LXVIIl, A, 1967-68, PART II.

rv

Het begrip verschilbasis, zoals gedefinieerd voor sommige cyclische groepen, kan worden gegenerahseerd voor sommige niet-abelse groepen, door het begrip quotiëntbasis in te voeren. Uitgaande van bepaalde niet-isomorfe groepen kunnen langs deze weg isomorfe projectieve meet-kunden worden geconstrueerd.

V

De bewering van E. T. COPSON dat een deelruimte A van een metrische ruimte dan en slechts dan onsamenhangend is als er twee disjuncte ge-sloten verzamelingen Fi en F2 bestaan zo dat A ^ Fi<uF2,Ar^Fi ^ 0 en Ar\F2¥'0, is wat het 'slechts dan' betreft, onjuist.

E. T. COPSON, Metric Spaces, Cambridge Uni-versity Press, 1968.

VI

De toenemende mate waarin de axiomatiek via de wiskunde ook het wiskunde-onderwijs overheerst, is geen onverdeeld succes.

VII

De afsluiting van de Oosterschelde is ter beveiliging van de aanüggende eilanden tegen stormvloeden de enig juiste oplossing.

VIII

Het feit dat sommige Nederlanders het Afrikaans beschouwen als een onvolwaardig dialect van het Nederlands, is voor een groot deel het ge-volg van hun geringe kennis van de Afrikaanse taal.

IX

Als een normcommissie, belast met de samenstelling van de Nederlandse nomenclatuur van bepaalde vakgebieden, het noodzakelijk oordeelt een Engelse vakterm te vervangen door een geschikte Nederlandse en daarin niet slaagt, verdient het aanbeveling dat zij ook kennis neemt van des-betreffende Zuid-Afrikaanse vakwoordenlijsten, ten einde daarin inspi-ratie te zoeken ter vervaardiging van de gewenste Nederlandse term.

X

De ogenschijnlijke economische integratie van Zuid-Afrika en zijn buur-staten wekt ten onrechte de indruk dat een gemeenschappelijke markt naar het voorbeeld van de Europese Economische Gemeenschap in de naaste toekomst in zuidelijk Afrika realiseerbaar is.