Artinian cofinite modules over complete Noetherian local rings

Let (R, m) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R-module. In this paper it is shown that if p is a prime ideal of R such that dim R/p = 1 and (0:M p) is not finitely generated and for each i ? 2 the R-module Ext _R ⁱ (M,R/p) is of finite length, then the R-module Ext _R ¹ (M, R/p) is not of finite length. Using this result, it is shown that for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext _R ⁱ (N,M) are of finite length, if and only if, for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext _R ⁱ (M,N) are of finite length.     

Some results on the cofiniteness of local cohomology modules

Let R be a commutative Noetherian ring, a an ideal of R, M an R-module and t a non-negative integer. In this paper we show that the class of minimax modules includes the class of AF modules. The main result is that if the R-module Ext _R ^t (R/a,M) is finite (finitely generated), H _a ⁱ (M) is a-cofinite for all i < t and H _a ^t (M) is minimax then H _a ^t (M) is a-cofinite. As a consequence we show that if M and N are finite R-modules and H _a ⁱ (N) is minimax for all i < t then the set of associated prime ideals of the generalized local cohomology module H _a ^t (M,N) is finite.     

Some results on the local cohomology of minimax modules

Let R be a commutative Noetherian ring with identity and I an ideal of R. It is shown that, if M is a non-zero minimax R-module such that dim Supp H _I ⁱ (M) ? 1 for all i, then the R-module H _I ⁱ (M) is I-cominimax for all i. In fact, H _I ⁱ (M) is I-cofinite for all i ? 1. Also, we prove that for a weakly Laskerian R-module M, if R is local and t is a non-negative integer such that dim Supp H _I ⁱ (M) ? 2 for all i < t, then Ext _R ^j (R/I,H _I ⁱ (M)) and Hom _R (R/I,H _I ^t (M)) are weakly Laskerian for all i < t and all j ? 0. As a consequence, the set of associated primes of H _I ⁱ (M) is finite for all i ? 0, whenever dim R/I ? 2 and M is weakly Laskerian.     

Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N?=?IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M)?=?{PM?|?P?∈?Spec(R) and P???M ^⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.     

A generalization of the finiteness problem in local cohomology modules

Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H _a ⁱ (N) is a weakly Laskerian R-module for all i < t, then Hom _R (R/a, H _a ^t (M, N)) is weakly Laskerian R-module. Also, we prove that Ext _R ⁱ (R/a, H _a ^t )) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp _R (H _a ⁱ (N)) is a finite set for all i < t, then Ext _R ⁱ (R/a, H _a ^t (N)) is weakly Laskerian R-module for all i = 0, 1.     

Wakamatsu tilting modules with finite injective dimension

Let R be a left Noetherian ring, S a right Noetherian ring and _R ω a Wakamatsu tilting module with S = End( _R ω). We introduce the notion of the ω-torsionfree dimension of finitely generated R-modules and give some criteria for computing it. For any n ? 0, we prove that l.id _R (ω) = r.id _S (ω) ? n if and only if every finitely generated left R-module and every finitely generated right S-module have ω-torsionfree dimension at most n, if and only if every finitely generated left R-module (or right S-module) has generalized Gorenstein dimension at most n. Then some examples and applications are given.     

On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

Simple-Direct-Projective Modules

In this paper, we introduce and study the dual notion of simple-direct-injective modules. Namely, a right R-module M is called simple-direct-projective if, whenever A and B are submodules of M with B simple and M/A ? B ?^⊕M, then A ?^⊕M. Several characterizations of simple-direct-projective modules are provided and used to describe some well-known classes of rings. For example, it is shown that a ring R is artinian and serial with J²(R) = 0 if and only if every simple-direct-projective right R-module is quasi-projective if and only if every simple-direct-projective right R -module is a D3-module. It is also shown that a ring R is uniserial with J²(R) = 0 if and only if every simple-direct-projective right R-module is a C3-module if and only if every simple-direct-injective right R -module is a D3-module.     

Some remarks on multiplication ideals,ii

Let R bea commutative ring with identity. An R-module (ideal of R) A is called a multiplication module (ideal) if for each submodule N of A there exists an ideal I of R with N = I A. We give several characterizations of multiplication modules. Using the method of idealization we show how to reduce questions concerning multiplication modules to multiplication ideals. For example, we show that if S is a commutative R-algebra and ψ: M→an R-module homomorphism where M is a multiplication R-module and N is an S-module, then Sψ(M) is a multiplication S-module.     

Correspondences of Coclosed Submodules

We establish an order-preserving bijective correspondence between the sets of coclosed elements of some bounded lattices related by suitable Galois connections. As an application, we deduce that if M is a finitely generated quasi-projective left R-module with S = End _R (M) and N is an M-generated left R-module, then there exists an order-preserving bijective correspondence between the sets of coclosed left R-submodules of N and coclosed left S-submodules of Hom _R (M, N).     

Complete cohomological functors on groups

If Λ is a ring and A is a Λ-module, then a terminal completion of Ext^1_Λ(A, ) is shown to exist if, and only if, Ext^j_Λ(A, P)=0 for all projective Λ-modules P and all sufficiently large j. Such a terminal completion exists for every A if, and only if, the supremum of the injective lengths of all projective Λ-modules, silp Λ, is finite. Analogous results hold for Ext^1_Λ(,A) and involve spli Λ, the supremum of the projective lengths of the injective Λ-modules. When Λ is an integral group ring $\text{Z}$G, spli$\text{Z}$G is finite implies silp $\text{Z}$G is finite. Also the finiteness of spli is preserved under group extensions. If G is a countable soluble group, the spli $\text{Z}$G is finite if, and only if, the Hirsch number of G is finite.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

Sequentially Cohen-Macaulay Modules Under Base Change

ABSTRACT

Assume that ?:(R, ± 𝔪) → (S, ± 𝔫) is a local flat homomorphism between commutative Noetherian local rings R and S. Let M be a finitely generated R-module. We investigate the ascent and descent of sequentially Cohen-Macaulay properties between the R-module M and the S-module M ? _R  S.     

k-torsionless modules with finite Gorenstein dimension

Let R be a commutative Noetherian ring. It is shown that the finitely generated R-module M with finite Gorenstein dimension is reflexive if and only if M _p is reflexive for p ∈ Spec(R) with depth(R _p) ? 1, and $G - {\dim _{{R_p}}}$ (M _p) ? depth(R _p) ? 2 for p ∈ Spec(R) with depth(R _p) ? 2. This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for n ? 2 we give a characterization of n-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown that every R-module has a k-torsionless cover provided R is a k-Gorenstein ring.     

Subnormal weighted translation semigroups

A weighted translation semigroup {S_t} on L²($\text{R}$₊) is defined by $\text{(S}{}_{\mathrm{t}}\text{f)(x) =}\text{(φ(x)}\text{φ(x ? t)}\text{)f(x ? t)}$ for x ? t and 0 otherwise, where φ is a continuous nonzero scalar-valued function on $\text{R}$₊. It is shown that {S_t} is subnormal if and only if φ² is the product of an exponential function and the Laplace-Stieltjes transform of an increasing function of total variation one. A necessary and sufficient condition for similarity of weighted translation semigroups is developed.     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

Properties of a certain product of submodules

Let R be a commutative ring with identity, let M be an R-module, and let K ₁, . . . ,K _n be submodules of M: We construct an algebraic object called the product of K ₁, . . . ,K _n : This structure is equipped with appropriate operations to get an R(M)-module. It is shown that the R(M)-module M ⁿ   = M . . .M and the R-module M inherit some of the most important properties of each other. Thus, it is shown that M is a projective (flat) R-module if and only if M ⁿ is a projective (flat) R(M)-module.     

Artinness of Generalized Macaulay–Northcott Modules

Let R be an associated ring not necessarily with identity, M a left R-module having the property (F), and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. It is shown that the module [M ^S,≤] consisting of generalized inverse polynomials over M is an artinian left [[R ^S,≤]]-module if and only if M is an artinian left R-module.     

Co-Hopfian Modules of Generalized Inverse Polynomials

Let R be an associative ring not necessarily possessing an identity and (S, ≤) a strictly totally ordered monoid which is also artinian and satisfies that 0 ≤s for any s∈S. Assume that M is a left R-module having propertiy (F). It is shown that M is a co-Hopfian left R-module if and only if [M ^{S , ≤}] is a co-Hopfian left [[R ^{S , ≤}]]-module. Received October 14, 1998, Accepted October 15, 1999