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Associated Prime Submodules of a Multiplication Module

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Associated Prime Submodules of a Multiplication Module Honam Mathematical J. 39 (2017), No. 2, pp. 275{296 https://doi.org/10.5831/HMJ.2017.39.2.275 ASSOCIATED PRIME SUBMODULES OF A MULTIPLICATION MODULE Sang Cheol Lee, Yeong Moo Song∗, and Rezvan Varmazyar Abstract. All rings considered here are commutative rings with identity and all modules considered here are unital left modules. A submodule N of an R-module M is said to be extended to M if N = aM for some ideal a of R and it is said to be fully invariant if '(L) ⊆ L for every ' 2 End(M). An R-module M is called a [resp., fully invariant] multiplication module if every [resp., fully invariant] submodule is extended to M. The class of fully invariant multipli- cation modules is bigger than the class of multiplication modules. We deal with prime submodules and associated prime submodules of fully invariant multiplication modules. In particular, when M is a nonzero faithful multiplication module over a Noetherian ring, we characterize the zero-divisors of M in terms of the associated prime submodules, and we show that the set Aps(M) of associated prime submodules of M determines the set ZdvM (M) of zero-dvisors of M and the support Supp(M) of M. 1. Introduction Every ring considered in this paper is a commutative ring with iden- tity and every module considered is a unital left module. A prime number in the ring of integers is generalized to a prime ideal in a ring. Further- more a prime ideal in a ring is generalized to a prime submodule in a module. A number of authors have studied for the subject for a couple of decades and many results have been given to explore the nature of prime submodules and related concepts. In this paper, we will focus on one of the most useful and well-known types of prime submodules mentioned, the associated prime submodules. Received April 3, 2017. Accepted May 10, 2017. 2010 Mathematics Subject Classification. 13E05, 14A05, 13C99, 13C10. Key words and phrases. primes, associated primes, multiplication modules, fully invariant multiplication modules. The second author∗ was partially supported by Sunchon National University Research Fund in 2016. 276 Sang Cheol Lee, Yeong Moo Song ∗, and Rezvan Varmazyar A submodule N of an R-module M is said to be extended to M if N = aM for some ideal a of R and it is said to be fully invariant if '(N) ⊆ N for every ' 2 End(M). An R-module M is called a multiplication module if each submodule N of M is extended. An R- module M is called a fully invariant multiplication module if each fully invariant submodule N of M is extended (see [17].) As Patrick F. Smith mentioned in the paper [17], the study of multi- plication modules is due to [11]. A number of authors have studied the subject since then and many results have given to explore the nature of multiplication modules (see, for example, [16, p.180 " 5].) In the paper [17], he considers the notion of fully invariant multiplication modules which is a natural generalization of multiplication modules. \The idea behind the paper was that the class of multiplication modules is quite small and much smaller than the class of fully invariant multiplication modules. However the bigger class shares some of the properties of the smaller one." That is what he remembers. Every multiplication module is a fully invariant multiplication mod- ule. However, not every fully invariant multiplication module is a mul- tiplication module. In section 2, we find the six conditions under which every fully invariant multiplication module is a multiplication module (see Theorem 2.3.) Assume that M is a finitely generated module over a Noetherian ring. If every prime submodule is extended and for every submodule N of M which is not prime the ideal (N :R M) is prime, then it is shown that M is a fully invariant multiplication module (see Theorem 2.6.) Now assume that M is a fully invariant multiplica- tion module. We introduce the set X of prime submodules of M to a Zariski topology T . The resulting topological space (X; T ) is called the prime spectrum of M, denoting Spec(M). And then we study the basic properties of Spec(M) and the relative topological space (Y; TY ). Here Y = fP 2 X j (P : M) is maximalg. If M is a finitely generated multiplication module over a ring R, then it is shown that Spec(M) is homeomorphic to Spec(R=AnnR(M)) (see Theorem 2.22.) and Y is homeomorphic to the maximal spectrum Ω(R=Ann(M)) of R=Ann(M) (see Corollary 2.23.) For any R-module M, the support of M is defined to be Supp(M) = fp 2 Spec(R) j Mp 6= 0g. A prime ideal p of R is said to be associated to M if there exists an injective R-linear map R=p ! M. The support of M and the set Ass(M) of associated primes to M are both the subsets of Spec(R) and they are both defined from Ann(M). It is shown that Ass(M) ⊆ Supp(M) (see Corollary 3.3.) A prime submodule P of an Associated Prime Submodules of a Multiplication Module 277 R-module M is said to be associated to M if the prime ideal (P :R M) of R is associated to M. If M is a finitely generated multiplication module over a ring R, then we show in Theorem 3.9 that the set Aps(M) of associated primes to M is Aps(M) = Spec(M) \ fAnnM (m) j m 2 Mg and in Theorem 3.12 Aps(M) \ Y = Spec(M) \ fM(m) j m 2 Ass(M) \ Ω(R)g: Let R be any ring and let M be an R-module. An element r of R is called a zero-divisor on M if rm = 0 for some nonzero m of M. The set of zero-divisors on M will be written ZdvR(M). If M is a nonzero finitely generated module over a Noetherain ring, then it is shown in Theorem 4.3 that jAss(M)j < 1; hence [ ZdvR(M) = p: p2Ass(M) jAss(M)j<1 Let R be any ring and let M be an R-module. An element m of M is called a zero-divisor of M if am = 0 for some nonzero ideal a of R. (Notice that an element m of M is called a singular element if em = 0 for some nonzero essential ideal e of R.) The set of zero-divisiors of M will be written ZdvM (M). This is not a submodule of M in general. A ring R is said to be von Neumann regular if for each a 2 R there exists a b 2 R such that a = aba. If M is a fully invariant multiplication module and if R is a von Neumann regular ring, then it is shown in Proposition 4.5 that ZdvR(R)M = hZdvM (M)i. Finally, we show that the set Aps(M) of associated prime submodules of an R-module M determine the set ZdvM (M) of zero-divisors of M and Supp(M). More specifically, assume that M is a nonzero faithful multiplication module over a Noetherian ring R. Then it is shown in Theorem 4.9 that jAps(M)j < 1; hence [ ZdvM (M) = P: P 2Aps(M) jAps(M)j<1 Moreover if : Spec(M) ! Spec(R) defined by (P ) = (P : M) is surjective, then it is shown in Corollary 4.14 that −1 [ (Supp(M)) = fQg; Q2Aps(M) jAps(M)j<1 hence −1(Supp(M)) is closed. For undefined terms see [4], [13] and [14]. 278 Sang Cheol Lee, Yeong Moo Song ∗, and Rezvan Varmazyar 2. Prime Spectra of Fully Invariant Multiplication Modules In this section, we will discuss that prime submodules play a critical role in fully invariant multiplication modules like prime ideals do in commutative rings. A submodule L of an R-module M is called fully invariant if '(L) ⊆ L for every ' 2 End(M) (see [17].) The zero submodule and M itself are fully invariant. Any submodule generated by the union of fully invariant submodules is fully invariant, and the intersection of fully invariant sub- modules is fully invariant. A submodule N of an R-module M is said to be extended to M if there exists an ideal a of R such that N = aM. Lemma 2.1. Let R be a ring and let M be an R-module. Then every extended submodule of M is fully invariant. Not every fully invariant submodule of an R-module is extended. For example, in the paragraph just prior to [17, Proposition 2.3], Patrick F. Smith constructed an example of a Z-module N such that the socle L of N is a fully invariant submodule of N, but N is not extended. Every multiplication module is a fully invariant multiplication mod- ule. However, not every fully invariant multiplication module is a mul- tiplication module. The example of this is given below. Example 2.2. Let R be any ring. Consider a free R-module F of rank ≥ 2. Then by [16, Lemma 1.3 and Example 2.8] F is not a multiplication module, but by [17, Corollary 2.10] it is a fully invariant multiplication module. In view of this, it is natural for us to have a question\under what conditions is every fully invariant multiplication module a multiplication module?" To answer this question, let us consider for each element a 2 R the map 'a : M ! M defined by 'a(m) = am, where m 2 M.
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