Systems of ideals in commutative rings

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYC'ZNE XXV (1985)

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(Bydgoszcz)

Systems of ideals in commutative rings

Introduction. Additional structures in a commutative ring R such as a derivation, a higher derivation or a grading, distinguish in the natural way, in the set I(R) of all ideals in R, some subset M of ideals invariant under a derivation, or a higher derivation, or the subset of homogenous ideals. In these cases, the sets M satisfy properties A1-A8 listed below. We set these properties as the characterisation for the notion of a system of ideals.

The main purpose of this paper is to show that we can unify proofs of several well-known theorems on differential ideals or homogeneous ideals (i.e., the theorem on primary decomposition) by proving corresponding theorems on the system of ideals. Moreover, we study relations between ideals of a system and prime ideals in R.

1. The category of systems. Throughout this paper all rings are com­

mutative with identity and all ring homomojphisms preserve the identity.

For any ring R, by I(R) will be denoted the set of all ideals in a ring R.

If E a R is any subset of R, then r(E) will denote the radical of the ideal (£). Let A e M, and let Г be a multiplicatively closed subset of R ; then A T is the ideal consisting of all b e R , such that there exists t e T with b - t e A . If x e R , then A x denotes A T with T = (1, x, x 2, ...}

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1.1. A system o f ideals (shortly: a system) is a pair (R, M), where Я is a ring and M is a set of ideals of R, satisfying the following conditions:

Al. The ring R is an element of M.

A2. The intersection of any set of elements of M is an element of M.

A3. The union of any non-empty set totally ordered by inclusion of elements of M is an element of M.

A4. The null ideal belongs to M.

A5. If A, В belong to M, then A + B belongs to M.

A6. If A, В belong to M, then A -В belongs to M.

A7. If A, В belong to M, then A: В belongs to M.

A8. If A belongs to M, and x is any element of R, then Ax belongs to M.

If (R , M ) is a system, then elements of M are called M-ideals.

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1.2. If (R , M) and (S , N) are systems of ideals, then a ring homomorphism f: R - > S will be called a morphism of systems if

(1) The inverse image of any N-ideal is an M-ideal.

(2) The ideal generated in S by the image of any M-ideal is an N-ideal.

Systems with morphisms form a category.

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1. We can obtain a system of ideals of a ring R from:

(1) A derivation d of R (or of any set of derivations) as the set of ideals I such that d(I) cz 1 (see [3], [5]).

(2) A higher derivation (see [1], [2]) D = {<5„} of R as the set of ideals / such that 0„{I) cz / for n ^ 0.

(3) A grading of R as the set of homogeneous ideals.

It is easy to see that any homomorphism of rings preserving derivations, higher derivations or a grading is a morphism of systems.

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2. Let R be a ring; then (R, M), where M — I(R), is a system.

(R, (0, R}) is a system if and only if 0 is a primary ideal of R.

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3. Let (R, M) be a system and let A e M . Then there exists the unique set M/A of ideals in R/A such that (R / A , M / A ) is a system and the natural homomorphism ry. R -> R / A , induces a morphism of systems. The system (R/A, M / A ) will be called the quotient system.

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4. Let (R, M) be a system and S a multiplicatively closed set in R. There exists the unique set S _ 1 M of ideals in S " 1 R such that (S~1 R, S ~ 1 M) is a system, and the natural homomorphism /: R - > S ~ 1R induces a morphism of systems. The system (S-1 R, S'-1 M) will be called the system o f fractions of (R, M ) with respect to S.

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1.3 If £ is a subset of R, then we denote by [£ ] the smallest M-ideal containing E. If A is a ideal of R, then we denote by A # the greatest M-ideal contained in A.

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1.4. Let (R, M) be a system, E any subset of R and let A, B, A h where i e J , be ideals o f R. Then

(i) ([2], [5]). I f x e [ £ ] , then there exists a finite subset F cz E such that x e [ F ] .

(ii) ([8], Lemma 2) A is an M-ideal if and only if for every x e E the condition x e A implies [x] <= A.

(Hi) [A + B] = [A] + m ,

(iv) (П 4-)# = П (4)#»

i i

(v) (А :Б)# = (A :[B ]).

P ro o f. Properties (iii) and (iv) are obvious. We prove (v). Since (A : [B]) e M, we have (A : [B]) = ( A : [В]) # с (A : B) # . Conversely, if x e (A : B) # , then [x] c (A:B)# a (A:B) and B c=(T:[x]), so that [В] с (A:[x]), i.e., х е(Л :[В ]).

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1.5. Let f : (R , M )—►(£, N) be a morphism of systems. Then (i) For every A e l ( R ) , we have:

(a) S/([A]) = I f (A)}, (b) S f ( A , ) a ( S f ( A ) ) t . (ii) For every B e I ( S ) , we have:

(а) [ / _1(B)] c / - 1^ ] ) , (b ) / - 1(B#) = / - 1(B)# .

P ro o f. We prove, (ii) (b), proofs of other conditions are similar. Because / - 1 (B#) e M and / “ 1 (B#) c=f ~ 1(B) it follows that f ~ 1 (B#) = / “ 1 (B#) # d / _1(B)# . Conversely, since S f ( f ~ 1 (B)#)e JV and S / ( / - 1 (B#)) c S f f ' l (B) c= В so S / ( / _1(B)#) c B # and hence we have / - 1 (Я)# c : / - 1 ( S f ( f ~ 1 (B)#)) c f ~ l (B*).

We say that an M-ideal A in a system (R, M) has an M-basis if there exists a finite subset E a R such that A = [£ ]. We have the following

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1.6. I f an M-ideal [E] in a system (R, M) has an M-basis, then there exists a finite subset F c= E such that [F ] = [£ ].

P ro o f. The ideal [F ] has an M-basis, so [£ ] = [ jcj , x2, x j for some x 1? x 2, ..., x ne R . By Proposition 1.4 (1) it follows that there exist finite subsets F, с E (i = 1, 2, ..., n) such that xfe [ F (] for i = 1, 2, ..., n.

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Let F = ( J F f; then [F ] = [x l9 x2, x3, ..., x j = [ x J + C x ^ ... +[x„]

c= [F j] + .l.~ V [ F J = [F , и F2 u ... u F J c [F ] <= [F ], therefore [F ] = [F ].

A system (R, M) will be called a Noetherian system if every M-ideal has an M-basis. This condition is equivalent to each of the following conditions:

(1) M-ideals satisfy the ascending chain condition.

(2) Every non-empty set of M-ideals has a maximal element.

2. Prime, M-prime and primitive ideals. The modification of well-known characteristic properties of prime ideals lead to the following definitions. Let (R, M) be a system.

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2.1. An M-ideal P is primitive if there exists a multiplicatively closed subset S of R such that P n S = 0 , and P is maximal among M-ideals of R disjoint from 5. The set of primitive M-ideals will be denoted by B(R, M).

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2.2. An M-ideal P is an M-prime ideal if for M-ideals A, B,

the condition A B с P, implies A с P or В a P. The set of all M-prime

ideals of (R , M ) will be denoted by C(R, M). Moreover, by A(R, M) will be denoted the set of all prime ideals of R, which belong to M.

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2.3. Let (R , M) be a system and A e M . Following conditions are equivalent:

(1) A is a primitive ideal in (R, M).

(2) A is a primary ideal and A — r(A)# . (3) r(A) is a prime ideal and А = г (Л ) # .

(4) There exists a prime ideal P in R such that A = P # .

The proof is similar to the proof of the analogous theorem for differ­

ential rings ([4], Proposition 2.2).

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2.4. I f {А{}1е1 is a subset o f B(R, M ) (resp. C(R, M)), totally ordered by inclusion, then f) A t belongs to B(R, M) (resp. C(R, M)).

ie/

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2.5. Let f : (R, M) -*■(£, N) be a morphism of systems. Then:

(1) I f В is a primitive ideal in (S, N), then f ~ 1 (B) is a primitive ideal in (R, M).

(2) I f Q is N -prime, then f ~ 1(Q) is M-prime.

Relations between the sets A(R, M), B(R, M), C(R, M) yields

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2.6. For every system (R, M) we have:

A( R, M) c B(R, M) c C(R, M).

The idea of the proof is based upon [7] (using the standard trick). It is worth to notice that inclusions in the Theorem above are not equalities, in general. For example the proper inclusion A( R, M) B(R, M) holds for the system (R, M), where R = Z 2 [X ]/(X 2) and M — {0, R}.

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2.7. I f R is a Noetherian ring, then for every system (R, M) we have B(R, M) = C(R, M).

P ro o f. It is sufficient to prove that C(R, M) c B(R, M). Let P e C ( R , M). First we prove that P is primary. Let x y e P . Px belongs to M and since R is Noetherian Px = (P:xk) for some k e N .

Since y e ( P : x k) and (P:xk) e M , we have [у] c (P:xfc) and further xfce(P :[y]), so [xk] [ y ] c : P . But P is M-prime, therefore [xk] c: P or [у] с P. Finally P is primary.

If A is a maximal M-ideal containing P and disjoint from multi- plicatively closed set R\ r( P) , then A is a primitive ideal in (R, M). But P

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r(P), so because R is Noetherian, we have A n <= r(P)n <= P for some n e N . Since P is M-prime and A e M we get A

P. Therefore, P — A, i.e., P belongs to B(R, M).

3. Primary decompositions, associated M-prime ideals. In the first part,

we prove a generalization of the theorem on primary decomposition for

systems (R , M), where R is a Noetherian ring. Second part contains a description of M-prime ideals associated with an M-ideal. Let (R, M) be a system.

D efinition 3.1. An M-ideal Q different from R will be called M-primary if for any M-ideals A, В the condition A B c Q implies A c Q or Bn czQ for some natural n.

D efinition 3.2. An M-ideal A will be called M-irreducible if A is not an intersection of two M-ideals different from R.

Using methods analogous to methods used by Sato [9], A. Nowicki [6], for investigation of differential ideals, we can prove

L emma 3.3. I f R is a Noetherian ring, then every an M-primary ideal is primary, and every M-irreducible M-ideal is M-primary.

We obtain from the previous lemma, using standard methods:

T heorem 3.4. Let (R, M) be a system with a Noetherian ring R. Any M- ideal A posseses the irredundant primary decomposition A = Ql n __ n Qn such that Qt , Q2, ..., Q„ are M-ideals.

If A is an ideal of R, then we denote by Asse (/1) the set of all prime ideals in R associated with A.

D efinition 3.5. If A is an M-ideal, then every M-prime M-ideal of the form (A\ [x]), where хфА, will be called M-prime ideal associated with A.

We denote the set of all M-prime M-ideals associated with A by Ass(K M)(A) or by AssM(A).

T heorem 3.6. I f P = (0: [x]) for x Ф 0 is a maximal M-ideal in the set {(0: у), у ф 0}, then P is M-prime.

P ro o f. Suppose that p is not an M-prime ideal in (R, M). There exist ideals A, B e M such that А - В <= P, А ф P and В ф P. Let a e A \ P , b e B \ P . Then [a] • [b] c= P, and [а] ф P, [b] ф P. Since [x] [b] Ф 0 there exist Xi g [x], b1e [ b] su c h 'th a t х 1-Ь1 фО. We notice that [ b j x J c [ b j x

x [ x j] cz [b] • [x] c [x].

Hence we have P = (0: [x]) c (0: [b] [x]) c (0: [bj x x]). By the assum­

p tio n :^ : [x]) = (0: [bi x j ) . Therefore, P = (0: [x]) = (0: [b] [x]) but [a] [b]

c P = (0: [x]) so [a] • [b] • [x] = 0 that means [a] c (0: [b] [x]) = P, whence we have the contradiction with афР.

T heorem 3.7. Let (R, M) be a Noetherian system, and let A be an M-ideal. There exists only a finite number of maximal elements in the set {(A: [x]), хфА}.

P ro o f. Using a quotient system (see Example 3), we can reduce the

proof to the case A = 0. Suppose that P, = (0: [x,]), where i e l , x, Ф 0, are

all maximal elements. By Theorem 3.6 we have that P, are M-prime ideals

for i e l . Let E = {х(}1е/. Of course, [£ ] has an M-basis. By Lemma 1.6 we

have [E] = [x l5 ..., x„] for some n e N . For every i e l , x.-efxj, x 2, ..., x„],

so [x,] c= [x j] + [x 2] + ... + [x„]. Therefore,

Pi = (0: [хг]) =з(0: [ x j + ... + [x„]) = (0: [ x j ) n ... n (0: [x„])

= P r n P2 n ... n Pn =э P 1 ■ P2 ■... • Pn.

Since P ,e C (P , M), so P, => Pk, for some к = 1, 2, . . n, by maximality we have P, = Pk.

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3.8. Let (R , M ) be a system with an Noetherian ring R. Every M-ideal maximal among M-ideals disjoint from P \£ (P ) is an M-prime ideal of the form (0: [X]), where x Ф 0 (£(P) is the set of zero divisors of R ).

P ro o f. Let P be an M-ideal maximal among M-ideals disjoint from R\C(R). Since R \ £ ( R ) is a multiplicatively closed set, then P e B ( R , M).

By Theorem 2.6 we have that P e C ( R , M). Since R is a Noetherian ring, C(R) = {0: x 1) u ( 0 : x 2) u . . . u ( 0 : x„), where x l5 x2, ..., x„ are non-zero elements in R, and (0: x,), for / = 1 ,2 , ...,« , are prime ideals.

P c (0: x x) и ... u (0: x„) s o P c= (0: xk) for some к = 1, 2, ..., n. Therefore, P = P # c= (0: xfc) # . By Theorem 1.4 (V) we have (0: xk) # = (0: [xk]). Thus P Œ (0: [xk])c=C(P), where (0: [xk])e M , i.e., P = (0: [xk]).

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3.9. Let (R, M) be a system with a Noetherian ring R. Then:

(1) There exists only a finite number of M-ideals maximal among M-ideals disjoint from R\Ç(R).

(2) Every M-ideal maximal among M-ideals disjoint from R\C(R) is an M-prime ideal of the form (0: [x]), where x Ф 0 is such an element from R, that (0: x) is a prime ideal.

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3.10. Let (R, M) be a system with a Noetherian ring R. There exists only a finite number o f minimal M-prime ideals. Each minimal M-prime ideal is of the form (0: [x]), where x ф 0 is an element of R such that (0: x) is a prime ideal.

P ro o f. If 0 = Ai n A 2 Г)... n A„ is an irredundant primary decomposi­

tion of 0, then r(Ai) = (0: x,) is a prime ideal in R for / = 1, 2, ..., n. Let P be a minimal M-prime ideal in (R, M). By Theorem 2.7 it follows that P e B(R, M) and from Theorem 2.3 we know that P = r(P )# , and r(P) is a prime ideal in R. We have r(P) =) r(0) so P = r(P )# э г ( 0 ) # . Hence

Р з r(0)# = r{Al n . . . n 4 ) # = ( r ( A 1) n . . . n r { A „ ) ) #

= r ( T i) # n . . . n r ( T „ ) # = (0: Xj)# n (0 : x2) # n . . . n ( 0 : x„)#

= (0 :[x 1] ) n ( 0 : [x 2]) n ... n (0 : [ x j ) =э (0: [ x J H O : [x 2])...(0 : [x„]).

But P is an M-prime ideal so Р э (0: [xk]) for some к = 1, 2, .., n. Since

(0: xk) is a prime ideal so that (0: [xk]) = (0: xk) # is a primitive M-ideal. It

implies (0: [xt] ) e C ( iî, M). By the minimality of P we have P = (0: [xk]) for some к = 1, 2, . . n.

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3.11. I f (R , M) is a system with a Noetherian ring R, then Ass(/î>M)(0) is a finite set. Each ideal which belongs to Ass(*)M)(0) has a form (0: [y]), where (0: y)eAss*(0).

P ro o f. Let 0 = A 1 n A 2 n ... n A„ be an irredundant primary decomposi­

tion, where A lf A 2, ■.., A„e M. Let P, = r(A,) for г = 1 ,2 , . . . , n ; P,

= (0: X;), where xt- Ф 0 for i = 1, 2, ..., n. Suppose that P = (0: [x]) belongs to Ass(KM)(0). We prove that P is identical with one of the ideals (Pi)#, •••, (P„)#- By Theorems 2.7 and 2.3 we know that P e B ( R , M) so P

= r(P )# . We have now

P = r(P )# = (r(0: M ) ) # = (r(A! n . . . n A „ : [x]))#

= r({Ax: [ x ] ) n ...n ( A „ : [x]))# = (Pl n ... n P„)# = ( P j ) # n . . . n ( P „ ) # . Hence P c ( P ,.) # for every / = 1 ,2 , . . . , n and P з (P ^# (P2)# • • -(Pn) # . But P e C ( R , M) so P r» (P,)# for some / = 1, 2, ..., n. Finally P = (P,)# for some /.

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3.12. Let (R, M) be a system with a Noetherian ring R. I f 0 ~ r\ A 2 ... c\ A n is an irredundant primary decomposition, then

Ass(R M)(0) = {r(Ai)# , ..., г ( у 4„)# ] .

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3.13. Let (R, M) be a system with a Noetherian ring R, and for A e M, Ass*(A) = {Pl5 P 2, ...» P„}- Then

(1) A ss (R M)( v 4) = {(Pi)#, ..., (P„)#}.

(2) There exists only a finite number of minimal M-prime ideals, and each of them is a minimal element in Ass(*.M)(T).

(3) Every M-ideal maximal among M-ideals disjoint from P \( ( P ) , belongs to Ass(* M)(y4).

P ro o f. We reduce the proof to the case T = 0 and apply suitable theorems: ad (1) Corollary 3.12; ad (2) Theorem 3.10; ad (3) Corollary 3.9.

References

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Math. 24 (1972), 400-415.

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[3] I. K a p la n s k y , An introduction to differential algebra, Hermann, Paris 1957.

[4] W. F. K e ig h e r , Quasi-prime ideals in differential rings, Houst. J. Math. 4 (1978), 379-388.

[5] E. R. К о le h in, Differential algebra and algebraic groups, Academic Press, New York-London 1973.

[6] A. N o w ic k i, The primary decomposition of differential modules, Comment. Math. 21 (1979), 341-346.

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[8] —, Prime ideals structure in additive conservative systems Demonstratio Math. 17 (1984).

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