A Note on the Countable Union of Prime Submodules^{*}

M. R. Pournaki, M. Tousi

Abstract Let M be a finitely-generated module over a Noetherian ring R. Suppose 𝔞 is an ideal of R and let N = 𝔞M and 𝔟 = Ann(M/N). If 𝔟 ⊆ J(R), M is complete with respect to the 𝔟-adic topology, {P_{i}}_{i≥1} is a countable family of prime submodules of M, and x ∈ M, then x + N ⊆ _{⋃i≥1}P_{i} implies that x + N ⊆ P_{j} for some j ≥ 1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules.

Keywords: Prime submodule, 𝔞-Adic topology, Complete metric space. 2000 Mathematics Subject Classification: Primary 13C99, 13J10; Secondary 13E05.

1 Introduction

Let R be a commutative ring with identity. One of the fundamental cornerstones of commutative ring theory is the “prime avoidance” theorem, which states that, if 𝔭_{1}, …, 𝔭_{n} are prime ideals of R and 𝔞 is an ideal of R such that 𝔞 ⊆ ⋃_{i=1}^{n}𝔭_{i}, then 𝔞 ⊆ 𝔭_{j} for some 1 ≤ j ≤ n. In [4], the authors proved and used an extension of the following.

Theorem 1.1 Let R be a Noetherian local ring having maximal ideal 𝔪. Let {𝔭_{i}}_{i≥1} be a countable family of prime ideals of R, 𝔞 an ideal of R, and x ∈ R. If R is complete with respect to the 𝔪-adic topology, then x + 𝔞 ⊆ _{⋃i≥1}𝔭_{i} implies that x + 𝔞 ⊆ 𝔭_{j} for some j ≥ 1.

Consider M as a unitary left R-module. A submodule P of M is called prime submodule, if P ≠ M and for each r ∈ R and m ∈ M, rm ∈ P implies that r ∈ Ann(M/P) or m ∈ P. Of course, if we regard R as an R-module, then the concept of prime submodules is equivalent to the concept of prime ideals. Now it is natural to ask if the “prime avoidance” theorem is true for prime submodules. In fact the answer is affirmative as has been shown by Lu [2].

Theorem 1.2 Let M be an R-module and P_{1}, …, P_{n} be prime submodules of M. Let N be a submodule of M such that N ⊆ ⋃_{i=1}^{n}P_{i}. Also assume that Ann(M/P_{j}) ⊈ Ann(M/P_{k}) for j ≠ k. Then N ⊆ P_{j} for some 1 ≤ j ≤ n.

The main purpose of this paper is to generalize the above theorem to a countable family of prime submodules.

*This research was in part supported by a grant from IPM.

2 Main Result

Let R be a Noetherian ring, 𝔞 an ideal of R contained in the Jacobson radical of R, and M a finitely-generated R-module. The 𝔞-adic topology is metrizable and one suitable metric d may be given by requiring that, for x, y ∈ M with x and y different, d(x, y) = 1/2^{t}, where t is the greatest integer i such that x − y ∈ 𝔞^{i}M.

Now, we apply the Baire’s category theorem for deducing our main result. Note that the Baire’s category theorem states that a complete metric space is not the union of a countable family of nowhere dense subsets, and a subset of a metric space M is nowhere dense in M if and only if its closure has no interior point (see [1, page 38]). In the sequel, we denote the Jacobson radical of R by J(R).

Main Theorem 2.1 Let M be a finitely-generated module over a Noetherian ring R. Suppose 𝔞 is an ideal of R and let N = 𝔞M and 𝔟 = Ann(M/N). If 𝔟 ⊆ J(R), M is complete with respect to the 𝔟-adic topology, {P_{i}}_{i≥1} is a countable family of prime submodules of M, and x ∈ M, then x + N ⊆ _{⋃i≥1}P_{i} implies that x + N ⊆ P_{j} for some j ≥ 1.

We will need the following well-known result which is essentially due to Krull.

Lemma 2.2 (see [3, Theorem 8.10]) If M is a finitely-generated module over a Noetherian ring R and 𝔟 an ideal of R such that 𝔟 ⊆ J(R), then each submodule of M with respect to the 𝔟-adic topology is closed.

Proof of the Main Theorem.

By Lemma 2.2, each submodule of M and in particular N is closed with respect to the 𝔟-adic topology. One easily obtains that x + N is also a closed subset of M. So the completeness of M yields that x + N is a complete metric space. But by hypothesis we have x + N = _{⋃i≥1}((x + N) ∩ P_{i}) and therefore it turns out by Baire’s category theorem that, there is some j ≥ 1 for which the subset (x + N) ∩ P_{j} of x + N is not nowhere dense. But (x + N) ∩ P_{j} is closed in x + N and therefore the interior of (x + N) ∩ P_{j} in x + N is not empty. Let c be an element of the interior of (x + N) ∩ P_{j}. Thus there exists an open subset U of M such that c ∈ (x + N) ∩ U ⊆ (x + N) ∩ P_{j}. But c ∈ U implies that, there is k ∈ ℕ such that c + 𝔟^{k}M ⊆ U and hence c ∈ (x + N) ∩ (c + 𝔟^{k}M) ⊆ (x + N) ∩ P_{j}. It is easily deduced that N ∩ 𝔟^{k}M ⊆ P_{j}. But 𝔟^{k}M ⊆ N, so 𝔟^{k}M ⊆ P_{j} and 𝔟^{k} ⊆ Ann(M/P_{j}).

Since P_{j} is a prime submodule of M, it follows that Ann(M/P_{j}) is a prime ideal of R and so 𝔟 ⊆ Ann(M/P_{j}). Now we have 𝔞 ⊆ 𝔟 ⊆ Ann(M/P_{j}), and hence 𝔞M = N ⊆ P_{j}. But x − c ∈ N, so x + N ⊆ P_{j}, as required. □

Remark 2.3 i) Note that if M is a finitely-generated module over a Noetherian ring R, 𝔞_{1} ⊆ 𝔞_{2} ideals of R, and M complete with respect to the 𝔞_{2}-adic topology, then it is also complete with respect to the 𝔞_{1}-adic topology. This shows that, the main theorem is also true if we assume that M is complete in J(R)-adic topology.

ii) In the view of (i), it is clear that Theorem 1.1 is a particular case of the main theorem.

References

[1] E. T. Copson, “Metric Spaces”, Cambridge Tracts in Mathematics and Mathematical Physics, no. 57, Cambridge University Press, London, 1968.

[2] C. P. Lu, Unions of Prime Submodules, Houston J. Math. 23 (1997), no. 2, 203-213.

[3] H. Matsumura, “Commutative Ring Theory”, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989.

[4] R. Y. Sharp, P. Vámos, Baire’s Category Theorem and Prime Avoidance in Complete Local Rings, Arch. Math. (Basel) 44 (1985), no. 3, 243-248.

The Authors’ Addresses

M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran, and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@karun.ipm.ac.ir

M. Tousi, Department of Mathematics, Shahid Beheshti University, Evin, Tehran 19834, Iran, and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: mtousi@vax.ipm.ac.ir