The Envelope of Artinian Modules over Commutative Rings 1

International Journal of Algebra, Vol. 5, 2011, no. 13, 647 - 653

The Envelope of Artinian Modules over Commutative Rings N. Hadian Dehkordi and A. J. Taherizadeh Department of Mathematical Sciences and Computer Tarbiat Moallem University, 43 Mofateh Ave., Tehran, Iran [email protected],

[email protected]

Abstract The concept of covers of a (Notherian) module over a commutative (non-trivial) ring with identity introduced in [1]. J. Chaui in [1] presented some applications of the covers, such as Nakayama’s Lemma, Krull’s Theorem for Jaconson radicals, etc. Here, we introduce the dual of the above mentioned notion, which we call it the envelope of a module. Then, as application we generalize some known results of Artinian modules lover commutative rings) such as union theorem, Nakayama’s Lemma etc.

1

Introduction

Let R be a commutative ring (with non-zero identity) A an R-module and Max(R) be the set of all maximal ideals of R. According to [1] a cover of A is a subset Γ of Max(R) such that for any x(= 0) of A, there is M ∈ Γ such that (0 :R x) ⊆ M. Now let J := M ∈Γ M and 0 = A be a finitely generated R-module. Then JA = A, which generalizes the Nakayama’s Lemma. In n addition, if R is a Noetherian ring, then ∞ n=1 J A = 0 [1, Prop. 2.4]. This is

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N. Hadian Dehkordi and A. J. Taherizadeh

a generalization of Krull’s intersection Theorem. Next we mention two results from [1]. Proposition 1.1. Let Γ be a finite subset of Max(R) and M an R-module. n Set J = ∈Γ m. If n∈ J M = 0, then Γ is a cover of M. The next result asserts that for any finitely generated R-module over a Noetherian ring R, there exists a finite cover. Theorem 1.2. Let R be a Noetherian ring and M a finitely generated R n module. Then there is a finite subset Γ of Max(R) such that ∞ n=1 J M = 0, where J = ∈Γ m. In particular, Γ is a cover of M. In the first section, we introduce the dual notion of the cover of a module. We call it the envelope of a module. Then we deduce several results concerning properties of this concept. In particular, we prove the dual of above propositions. In section 3, we generalize a result of Kirby and two result of Tang [8, The. 2.3, 2.4]. These two latter theorems are as follows. Suppose A is an Artinian modules over the commutative ring (with non-zero identity) R and I is an ideal of R. Set G := n∈ Gn , G := n∈ Gn where Gn = Gn = 0 for n > 0 and Gn = (0 :A I −n+1 )/(0 :A I −n ), Gn = M/(0 :A I −n ) for n ≤ 0. Also n n suppose that GI (R) := ∞ n=0 I X (X is an indeterminate), is the Ress ring of I and GI (R) = ⊕I n /I n+1 is the associated graded ring of I. Then G(G ) is an Artinian GI (R)(GI (R)) module [8, Pro. 2.2]. Moreover, if I ⊆ JA (M) = Supp(M) and lR (0 :A I) < ∞ then i) K dimGI (R) G = K dimR (A), ⎧ ⎪ ⎨K dimR (A) if K dimR (A) = 0 ii) K dimGI (R) G = ⎪ ⎩K dimR (A) + 1 if K dimR (A) ≥ 1. (Here K dimR (A) stands for Krull dimension of the Artinian module A as defined in [7, P.269]). Now let T be an envelope of the Artinian module A and J =

∈T

m, then

The envelope of Artinian modules over commutative rings

649

with the above notation, we prove these two results whenever a is an ideal of R which is contained in J.

2

The Envelope

We start this section with some definitions. Definition 2.1. An R-module L is said to be cocyclic if L is a submodule of D (R) for some m ∈ Max(R). (For any maximal ideal m of R the functor Hom(−, E(R/m)) is denoted by D (−)). An R-module L is said to be finitely embedded if E(M) = E(R/m1 ) ⊕ E(R/m2 ) ⊕ . . . ⊕ E(R/mk ), where each mi is a maximal ideal of R. Definition 2.2. Let M be an R-module. A prime ideal p of R is called coassociated [resp. weakly coassociated] if there exists a cocyclic homomorphis image L of M with p = (0 :R L) [resp. p is minimal over (0 :R L)]. The set of coassociated primes of M is denoted by CoassR M (resp. Coass R M ). It is clear that Coass M = φ if M = 0. We now define the envelop of a module. Definition 2.3. Let M be an R-module. A subset T of Max(R) is called an envelope of M if for any non-zero cocylic quotient L of M, there is m ∈ T such that (0 :R L) ⊆ m. Lemma 2.4. Let M be an R-module and T a subset of Max(R). Suppose for any nonzero finitely embedded quotient L of M, there is m ∈ T such that (0 :R L) ⊆ m. Then T is an envelope of M. Lemma 2.5. Suppose M is an R-module and T ⊆ Max R is an envelope of M. If r ∈ ∈T m, then M = rM.

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R M/rM = φ. Let p ∈ Coass R M/rM. Proof. Assume that M = rM, then Coass There is a cocylic quotient L of M/rM with p is minimal over (0 :R L). Since L is a cocylic quotient of M, there is m ∈ T , such that (0 :R L) ⊆ m. Now we get r ∈ (0 : M/rM) ⊆ (0 :R L) ⊆ m, which is a contradiction. The following may be considered as the dual of Krull intersection Theorem for Artinian modules. Theorem 2.6. Let R be a semi-local Noetherian complete ring and a an ideal of R. Let A be an Artinian R-module. Then there is r ∈ a such that (1+r)A ⊆ Γ (A), where Γ (A) := n∈ (0 :A an ).

E(R/m), D(.) = Hom(., E). Let M = D(A). Since n n ∼ lim Γ (M) = − n Hom(R/a , A) and so D(Γ (M)) = lim n M/a M, it is easy → ← − to see that n∈ an M ∼ = D(A/Γa (A)). From the Krull intersection theorem, we deduce that there is r ∈ a such that (1 + r) n≥0 an M) = 0. Thus (1 + Proof. Let E =

∈Max R

r)D(A/Γ(A) ) = 0. Thus (1 + r)a ⊆ Γ (A). Note that AnnR L = AnnR D(L) for any R-module. Corollary 2.7. (i) With the same assumption as Theorem 2.6, if T is an envelope of A and J = ∈T m, then A = Γj (A). (ii) (0 :A a) = 0 if and only if A = 0. (for all a ⊆ J.) Lemma 2.8. Let A be an Artinian R-module. Then there is an envelop T of A such that T is a finite set. Such an envelope is called a finite envelope of A. Proof. Let A/B be a non-zero arbitrary cocyclic quotient of A and y ∈ R such that yA ⊆ B = A, so yA = A and by [4(2.6)(i)] y ∈ ∈Att(A) p. Thus (0 :R A/B) ⊆ ∈Att(A) p, and hence (0 :R A/B) ⊆ p for some p ∈ Att(A). Suppose Att(A) = {p1 , . . . , pt }. For each pi choose Mi ∈ Max(R) such that pi ⊆ Mi for i = 1, . . . , t. Then {M1 , . . . , Mt } is a finite envelope of A. Remark 2.9. (i) If A is an Artinian R-module and Supp(A) = {M1 , . . . , Ms }, then by the proof of (2.8) and [5, (2.6)] Supp(A) is an envelope of A.


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(ii) For any Artinian R-module A, and any envelop T of it, T ⊇ CoassR (A)∩ Max(R). (iii) If T is an envelope of the R-module A and B is a submodule of A, generally it is not true that T is an envelope of B as well. For instance, if T is an envelope of the ring R and T ⊂ Max(R), then for any M ∈ Max(R)\T , T is not an envelope of (0 :R M).

=

(iv) In general, if T is an envelope of the submodule B of the R-module A, T is not necessarily an envelope of A. For example, suppose that the ring R has at least two maximal ideal and M ∈ Max(R) such that (0 :R M) = 0. Then {M} is the only envelope of (0 :R M) and {M} is not an envelope the R-module R.

3

Some further applications

In this section we generalize three results. The first one is due to Kirby [3, (270] and the second and third one is due to Tang [8, The. 2.3,2.4]. Theorem 3.1. Let R be a semi-local Noetherian complete ring and A an Artinian R-module. Let T be an envelope of A and a an ideal of R such that n a⊆ ∈T m, l(0 :A a) < ∞. Then the degree of polynomial function l(0 :A a ) is exactly K dimR (A). Proof. By (2.7) A = ΓJ (A) and so by [4,(1.6)], the result follows. Remark 3.2. Let G, G, G (R), G (R) be the same as in introduction and T = {m1 , . . . , mt } be a finite envelope of A and J = ti=1 mi . Then by (2.7) A = Γj (A).

(1)

Set Mi := mi ⊕ a ⊕ a2 ⊕ . . . (Mi := mi ⊕ a/a/a2 ⊕ . . . ) for i = 1, 2, . . . , t; and put L1 := ti=1 Mi , L2 := ti=1 Mi . Then it is easy to deduce from (1) that ∞

n=1

(0

:G Ln1 )

= G and

∞

n=1

(0 :G Ln2 ) = G

(2)

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It follows from (2) and (1.8) that {M1 , . . . , Mt }({M1 , . . . , Mt }) is an envelope for G (G resp.). Theorem 3.3. With the same notation and assumptions as above, suppose a ⊆ tu=1 mi and l(0 :A a) < ∞. Then = K⎧dimR (A), and ⎪ ⎨K dim (A) ii) K dimG (R) (G ) = ⎪ ⎩K dim (A) + 1 i) K dimG

(R) (G)

if K dimR (A) = 0 if K dimR (A) > 0

Proof. By the above remark and (2.7) {M1 , . . . , Mt } and {M1 , . . . , Mt } are envelopes of G and G respectively. Now the result follows by the same argument as in [8. The. 23,2.4] but using (3.1) instead of [3.(2.7)]. Acknowledgment. The first author owe thanks to Professor K. DivaaniAazar for his invaluable help.

References [1] J. Chuai, The Covers of a Noetherian module, Pac. J. Math. (1), 173 (1996), 69-76. [2] K. Divanni-Azzar and M. Tousi, Some Remarks on Coassociated Primes, J. Korean Math. Soc. 36(1999), No. 5, 847-853. [3] D. Kirby, Artinian modules and Hilbert polynomials, Quart. J. Math. Oxford (2), 24(1973), 47-57. [4] D. Kirby, Dimension and length for Artinian modules, Quart, J. Math. Oxford (2), 41 (1990), 419-429. [5] I.G. Macdonald, Secondary representation of modules over a commutative ring, Symposia Mathematica, 11(1973), 23-43. [6] I. Nishitani, On the dual of Burch’s inequality, J. of Pure and Appl. Algebra, 96(1994), 147-156.


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[7] R.N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings, Quart. J. Math. Oxford (2), 26(1975), 269-273. [8] Z. Tang, On certain graded Artinian modules, Comm. Algebra, 21(1993), 255-268. [9] S. Yassemi, Coassociated Primes of Modules Over a Commutative Rings, Math. Scand., 80(1997), 175-187. Received: January, 2011