Nonzero R-module Research Articles

On F-Z-hollow and Z-semihollow modules

In this work, F-Z-hollow and Z-semihollow modules are defined as a generalization of F-hollow and semihollow modules, and some properties of these concepts of modules are investigated. Where an R-module E over a ring R is F-Z-hollow if every proper submodule of E is F-Z-small in E. And we say that a nonzero R-module E is Z-semihollow module if every proper finitely generated submodule of E Z-small submodule.

Decomposition of modules into modules whose annihilators are pseudo-irreducible

ABSTRACTA nonzero module M over a commutative ring R is said to have a complete comaximal decomposition if it can be written in the form , where the annihilators of the Ni’s are pairwise comaximal pseudo-irreducible ideals of R. In this paper, we show that a complete comaximal decomposition for an R-module is unique if it exists and a ring R is J-Noetherian if and only if every nonzero R-module has a complete comaximal decomposition. Also, we give a topological characterization of (finitely generated) comultiplication R-modules whose submodules have a complete comaximal decomposition and then we show that a commutative ring R is von Neumann regular if and only if every representable R-module can be written as a finite direct sum of homogeneous semi-simple R-modules.

Extension functors of cominimax modules

ABSTRACTLet R be a commutative Noetherian ring with identity and I be an ideal of R. Also, let M and N be two nonzero R-modules. We prove that the R-module is I-cominimax for all i≥0, whenever M is I-cominimax and N is finitely generated with dim N ≤ 2. Also, it is shown that the R-module is I-cominimax for all i ≥ 0, whenever N is finitely generated and M is I-cominimax with dim M ≤ 1. As an immediate consequence, we obtain that if M is a nonzero minimax R-module such that , then for each finitely generated R-module N, is I-cominimax for all i ≥ 0 and j ≥ 0. Moreover, we prove that if R is local, M is I-cominimax and N is finitely generated, then the R-module is I-weakly cofinite for all i ≥ 0, when one of the following statements holds:(i) dim N = 3 or(ii) dim M ≤ 2.

Zero-Divisors, Torsion Elements, and Unions of Annihilators

Let R be a commutative ring and M be a nonzero R-module. Now Z(M) = {r ∈ R | rm = 0 for some 0 ≠ m ∈ M} is a union of prime ideals of R and T(M) = {m ∈ M | rm = 0 for some 0 ≠ r ∈ R} is a union of prime submodules of M if M ≠ T(M). We investigate representations of Z(M) and T(M) as unions of primes each of which is a union of annihilators.

Minimaxness and Cofiniteness Properties of Local Cohomology Modules

Let R be a commutative Noetherian ring, I an ideal of R, and M a non-zero R-module. The purpose of this article is to introduce the notation of the nth finiteness dimension for all n ∈ ℕ0, and to prove the following results: (i) ; (ii) The R-modules are I-cofinite for all and for all minimax submodules N of , the R-modules are finitely generated, whenever is finite. This implies that if I has dimension one, then is I-cofinite for every i ≥ 0, which is a generalization of the main results of Delfino–Marley, Yoshida, and Bahmanpour–Naghipour. (iii) , whenever R is semilocal. (iv) The R-modules are weakly Laskerian for all j ≥ 0 and all , whenever (R, 𝔪) is a complete Noetherian local ring. Moreover, in this situation for all weakly Laskerian submodules N of , the R-modules are weakly Laskerian, whenever is finite. In addition, some examples about , for n = 0, 1, 2, 3, are included.

Second Modules Over Noncommutative Rings

Let R be an arbitrary ring. A nonzero unital right R-module M is called a second module if M and all its nonzero homomorphic images have the same annihilator in R. It is proved that if R is a ring such that R/P is a left bounded left Goldie ring for every prime ideal P of R, then a right R-module M is a second module if and only if Q = ann R (M) is a prime ideal of R and M is a divisible right (R/Q)-module. If a ring R satisfies the ascending chain condition on two-sided ideals, then every nonzero R-module has a nonzero homomorphic image which is a second module. Every nonzero Artinian module contains second submodules and there are only a finite number of maximal members in the collection of second submodules. If R is a ring and M is a nonzero right R-module such that M contains a proper submodule N with M/N a second module and M has finite hollow dimension n, for some positive integer n, then there exist a positive integer k ≤ n and prime ideals P i (1 ≤ i ≤ k) such that if L is a proper submodule of M with M/L a second module, then M/L has annihilator P i for some 1 ≤ i ≤ k. Every second submodule of an Artinian module is a finite sum of hollow second submodules.

Test Modules for Projectivity of Duals

Over a regular local ring R, given a nonzero R-module N and an arbitrary R-module M, we examine conditions for the freeness of the dual M*, in terms of vanishing of the groups, .

A Note on Dedekind and ZPI Modules

Let R be a domain. A non-zero R-module M is called a Dedekind module if every submodule N of M such that N ≠ M either is prime or has a prime factorization N=P1P2… PnN*, where P1, P2, … Pn are prime ideals of R and N* is a prime submodule in M. When R is a ring, a non-zero R-module M is called a ZPI module if every submodule N of M such that N ≠ M either is prime or has a prime factorization. The purpose of this paper is to introduce interesting and useful properties of Dedekind and ZPI modules.

IDEALS AND SUBMODULES OF MULTIPLICATION MODULES

Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = 1M. Let M be a non-zero multiplication R-module. Then we prove the following: (1) there exists a bijection: N(M)<TEX>$\bigcap$</TEX>V(ann<TEX>$\_{R}$</TEX>(M))<TEX>$\rightarrow$</TEX>Spec<TEX>$\_{R}$</TEX>(M) and in particular, there exists a bijection: N(M)<TEX>$\bigcap$</TEX>Max(R)<TEX>$\rightarrow$</TEX>Max<TEX>$\_{R}$</TEX>(M), (2) N(M) <TEX>$\bigcap$</TEX> V(ann<TEX>$\_{R}$</TEX>(M)) = Supp(M) <TEX>$\bigcap$</TEX> V(ann<TEX>$\_{R}$</TEX>(M)), and (3) for every ideal I of R, The ideal <TEX>$\theta$</TEX>(M) = <TEX>$\sum$</TEX><TEX>$\_{m</TEX>(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P <TEX>$\in$</TEX> Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P<TEX>$^{n}$</TEX>M<TEX>$\neq$</TEX>P<TEX>$^{n+1}$</TEX> for every positive integer n, then <TEX>$\bigcap$</TEX><TEX>$^{</TEX><TEX>$\_{n=1}$</TEX>(P<TEX>$^{n}$</TEX> + ann<TEX>$\_{R}$</TEX>(M)) <TEX>$\in$</TEX> V(ann<TEX>$\_{R}$</TEX>(M)) = Supp(M) <TEX>$\subseteq$</TEX> N(M).

Modules with many homomorphisms

It is proved that if R is a right FBN ring then a non-zero right R-module M has the property that Hom R ( M , N ) ≠ 0 for every non-zero submodule N of M if and only if Hom R ( M , R / P ) ≠ 0 for every associated prime ideal P of M. One consequence is that over a commutative Noetherian ring R, Hom R ( X , Y ) ≠ 0 for every non-zero projective R-module X and every non-zero submodule Y of X. In case R is a left Noetherian right FBN ring, then a non-zero finitely generated right R-module M has the property that Hom R ( M , N ) ≠ 0 for every non-zero submodule N of M if and only if the right ( R / P ) -module M / MP is not torsion for every associated prime ideal P of M. Finally, if R is a commutative Noetherian ring and M is an R-module such that Hom R ( M , R ) ≠ 0 then Hom R ( M , M ′ ) ≠ 0 for every non-zero R-module M ′ . It is shown that this result does not extend to prime Noetherian PI rings.

A theorem on artinian rings

In this short paper, we prove that a commutative Noetherian ring R is Artinian if and only if for any nonzero R-module M there is a nonzero linear map from M into F with F flat and if and only if there is a nonzero linear map from M into P with P projective. An example will show that this is not true for noncommutative rings. The motivation of the present discussion is partially from [I] where flat envelopes of modules were studied. It is also motivated by the study of embeddings of modules into projective modules.

A result by Kulikov that does not extend to modules over general valuation domains

Let R be a valuation domain. It is proved that every nonzero R-module contains a nonzero pure uniserial submodule if and only if R is rank-one discrete. It is a classical result in the theory of Abelian groups, due to Kulikov [5], that every Abelian p-group contains a pure cocyclic subgroup (which is necessarily a summand, cocyclic groups being pure-injective). Note that cocyclic p-groups are exactly the torsion uniserial Zp-modules, where 7p is the rank-one discrete valuation domain obtained by localizing the ring of the integers Z at the prime P. The goal of this paper is to show that this result is not extendable to valuation domains which are not rank-one discrete. Our main result is the following: Theorem. Let R be a valuation domain. Then every nonzero R-module contains a nonzero pure uniserial submodule if and only if R is rank-one discrete. The first contribution to the proof of this theorem was given in [2] (see also [3, X.4]), where an example of a cohesive R-module (i.e., a module without elements of limit height) with no nonzero uniserial pure submodules was given, under the assumption that the maximal ideal P of R is not principal. The second author showed in [6] that a cohesive R-module with the above property cannot exist if and only if J * RJ is a principal ideal of RJ, for each prime ideal J of R (equivalently, R is discrete and Spec(R) is well ordered by the opposite inclusion); note that the proof of the sufficiency was just an adaptation of the example given in [2]. The consequence of the above results is that, in order to complete the proof of the theorem, we must find a noncohesive R-module with no nonzero pure uniserial submodules, where R is any discrete valuation domain with a wellordered prime spectrum consisting of at least two nonzero ideals. In fact, we shall construct an R-module S, where R is as in the preceding Received by the editors October 16, 1989 and, in revised form, March 22, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 13C05, 13C12.

Finite dimensional hereditary algebras of wild representation type

Let R be a finite dimensional hereditary algebra. We are concerned with the problem of determining the indecomposable R-modules of finite length. This problem completely has been solved in the case when R is of finite or of tame representation type, but seems to be rather hopeless in the case of wild representation type. In this situation, the only known classes of modules are the socalled pre-projective and the pre-injective ones. The remaining indecomposable modules are called regular. In this paper, we want to initiate the study of the regular modules. The result we obtain seems to be rather surprising: we will show that the regular modules behave rather similar to modules over a serial algebra. In order to state the main theorem, we need the notion of an irreducible homomorphism, which was introduced by Auslander and Reiten [-3]. Let X and Y be two non-zero R-modules. A homomorphism f : X ~ Y is said to be irreducible, if it is neither a split monomorphism, nor a split epimorphism, and if for any factorisation X ~ I ~ Y o f f , either f is a split monomorphism o r f is a split epimorphism. Note that an irreducible homomorphism is always either a monomorphism or an epimorphism. A non-zero R-module S will be called quasisimple, ifS is regular, and there is no irreducible monomorphism of the form U ~ S with U non-zero. In this case, we will call the map 0 ~ S irreducible.