Envelopes, covers and semidualizing modules

2019 ◽  
Vol 18 (07) ◽  
pp. 1950137
Author(s):  
Lixin Mao
Keyword(s):  
Commutative Ring ◽  
Noetherian Rings ◽  
Artinian Rings ◽  
Injective Modules ◽  
Semidualizing Modules ◽  
Coherent Rings ◽  
The Relationship

Given an [Formula: see text]-module [Formula: see text] and a class of [Formula: see text]-modules [Formula: see text] over a commutative ring [Formula: see text], we investigate the relationship between the existence of [Formula: see text]-envelopes (respectively, [Formula: see text]-covers) and the existence of [Formula: see text]-envelopes or [Formula: see text]-envelopes (respectively, [Formula: see text]-covers or [Formula: see text]-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect rings and Artinian rings in terms of envelopes and covers by [Formula: see text]-projective, [Formula: see text]-flat, [Formula: see text]-injective and [Formula: see text]-[Formula: see text]-injective modules, where [Formula: see text] is a semidualizing [Formula: see text]-module.

Some remarks on Nonnil-coherent rings and ϕ-IF rings

2021 ◽  
pp. 2250211
Author(s):  
Wei Qi ◽  
Xiaolei Zhang
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Injective Module ◽  
Nilpotent Radical ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Rings

Let [Formula: see text] be a commutative ring. If the nilpotent radical [Formula: see text] of [Formula: see text] is a divided prime ideal, then [Formula: see text] is called a [Formula: see text]-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and [Formula: see text]-coherent rings introduced by Bacem and Ali [Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297–305], and then characterize nonnil-coherent rings in terms of [Formula: see text]-flat modules, nonnil-injective modules and nonnil-FP-injective modules. A [Formula: see text]-ring [Formula: see text] is called a [Formula: see text]-IF ring if any nonnil-injective module is [Formula: see text]-flat. We obtain some module-theoretic characterizations of [Formula: see text]-IF rings. Two examples are given to distinguish [Formula: see text]-IF rings and IF [Formula: see text]-rings.

Decomposition of totally transcendental modules

1980 ◽  
Vol 45 (1) ◽  
pp. 155-164 ◽  
Author(s):  
Steven Garavaglia
Keyword(s):  
Quantifier Elimination ◽  
Decomposition Theorem ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Artinian Modules ◽  
Injective Modules ◽  
Unique Decomposition ◽  
Special Cases ◽  
Decomposition Theorems ◽  
Coherent Rings

The main theorem of this paper states that if R is a ring and is a totally transcendental R-module, then has a unique decomposition as a direct sum of indecomposable R-modules. Natural examples of totally transcendental modules are injective modules over noetherian rings, artinian modules over commutative rings, projective modules over left-perfect, right-coherent rings, and arbitrary modules over Σ – α-gens rings. Therefore, our decomposition theorem yields as special cases the purely algebraic unique decomposition theorems for these four classes of modules due to Matlis; Warfield; Mueller, Eklof, and Sabbagh; and Shelah and Fisher. These results and a number of other corollaries about totally transcendental modules are covered in §1. In §2, I show how the results of § 1 can be used to give an improvement of Baur's classification of ω-categorical modules over countable rings. In §3, the decomposition theorem is used to study modules with quantifier elimination over noetherian rings.The goals of this section are to prove the decomposition theorem and to derive some of its immediate corollaries. I will begin with some notational conventions. R will denote a ring with an identity element. LR is the language of left R-modules described in [4, p. 251] and TR is the theory of left R-modules. “R-module” will mean “unital left R-module”. A formula will mean an LR-formula.

scholarly journals Secondary representations for injective modules over commutative Noetherian rings

1976 ◽  
Vol 20 (2) ◽  
pp. 143-151 ◽  
Author(s):  
Rodney Y. Sharp
Keyword(s):  
Representation Theory ◽  
Commutative Ring ◽  
Noetherian Rings ◽  
Primary Decomposition ◽  
Dual Theory ◽  
Injective Modules ◽  
Finite Sum ◽  
Secondary Representation

There have been several recent accounts of a theory dual to the well-known theory of primary decomposition for modules over a (non-trivial) commutative ring A with identity: see (4), (2) and (9). Here we shall follow Macdonald's terminology from (4) and refer to this dual theory as “ secondary representation theory ”. A secondary representation for an A-module M is an expression for M as a finite sum of secondary submodules; just as the zero submodule of a Noetherian A-module X has a primary decomposition in X, it turns out, as one would expect, that every Artinian A-module has a secondary representation.

SEMIDUALIZING MODULES AND RELATED MODULES

2011 ◽  
Vol 10 (06) ◽  
pp. 1261-1282 ◽  
Author(s):  
DONGDONG ZHANG ◽  
BAIYU OUYANG
Keyword(s):  
Noetherian Ring ◽  
Injective Modules ◽  
Bass Class ◽  
Derived Functors ◽  
Semidualizing Modules ◽  
Left And Right ◽  
Definition Of ◽  
Semidualizing Bimodule ◽  
Coherent Rings

In this paper, we prove that the Bass class [Formula: see text] with respect to a semidualizing bimodule C contains all FP-injective S-modules. We introduce the definition of C-FP-injective modules, and give some characterizations of right coherent rings in terms of the C-flat S-modules and C-FP-injective S op -modules. We discuss when every S-module has an C-flat preenvelope which is epic (or monic). In addition, we investigate the left and right [Formula: see text]-resolutions of R-modules by left derived functors Ext n(-, -) over a left Noetherian ring S. As applications, some new characterizations of left perfect rings are induced by these modules associated with C. A few classical results of these rings are obtained as corollaries.

Finitely projective modules with respect to a semidualizing module

2019 ◽  
Vol 18 (03) ◽  
pp. 1950049
Author(s):  
Lixin Mao
Keyword(s):  
Commutative Ring ◽  
Projective Modules ◽  
Artinian Rings ◽  
Semidualizing Module ◽  
Coherent Rings

Let [Formula: see text] be a commutative ring. We define and study [Formula: see text]-projective modules with respect to a semidualizing [Formula: see text]-module [Formula: see text], which are called [Formula: see text]–[Formula: see text]-projective modules. As consequences, we characterize several rings such as [Formula: see text]-coherent rings and Artinian rings using [Formula: see text]–[Formula: see text]-projective modules. Some known results are extended.

scholarly journals Artinian bimodule with quasi-Frobenius bimodule of translations

2019 ◽  
Vol 29 (2) ◽  
pp. 103-119
Author(s):  
Aleksandr A. Nechaev ◽  
Vadim N. Tsypyschev
Keyword(s):  
Commutative Ring ◽  
Recurrent Sequence ◽  
Artinian Rings ◽  
Linear Recurrent Sequence ◽  
Left And Right ◽  
Additional Assumptions

Abstract The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well.

On commutative rings with uniserial dimension

2014 ◽  
Vol 14 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. Ghorbani ◽  
Z. Nazemian
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Artinian Rings ◽  
Uniform Dimension ◽  
Uniserial Modules

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

2-Visible Submodules and Fully 2-Visible Modules

2019 ◽  
pp. 1-6
Author(s):  
Mahmood S. Fiadh ◽  
Wafaa H. Hanoon
Keyword(s):  
Commutative Ring ◽  
Nonzero Ideal ◽  
The Relationship

Let be a -module, T is a commutative ring with identity and be a proper submodule of . In this paper we introduce the concepts of 2-visible submodules and fully 2-visible modules as a generalizations of visible submodules and fully visible modules resp., where is said to be 2-visible whenever for every nonzero ideal of and A -module is called fully 2-visible if for any proper submodule of it is 2-visible.Study some of the properties of these concepts also discuss the relationship 2-visible submodules and fully 2-visible modules with 2-pure submoules and other related submodules and modules resp. are given.

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Noetherian Ring ◽  
Direct Limit ◽  
Injective Module ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Injective Modules ◽  
Pure Submodules ◽  
Covering Class ◽  
Gorenstein Injective ◽  
Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

Definability problems for modules and rings

1971 ◽  
Vol 36 (4) ◽  
pp. 623-649 ◽  
Author(s):  
Gabriel Sabbagh ◽  
Paul Eklof
Keyword(s):  
Ring Theory ◽  
Noetherian Rings ◽  
Fixed Ring ◽  
Injective Modules ◽  
Free Left ◽  
Right Noetherian Rings ◽  
The Right

This paper is concerned with questions of the following kind: let L be a language of the form Lαω and let be a class of modules over a fixed ring or a class of rings; is it possible to define in L? We will be mainly interested in the cases where L is Lωω or L∞ω and is a familiar class in homologic algebra or ring theory.In Part I we characterize the rings Λ such that the class of free (respectively projective, respectively flat) left Λ-modules is elementary. In [12] we solved the corresponding problems for injective modules; here we show that the class of injective Λ-modules is definable in L∞ω if and only if it is elementary. Moreover we identify the right noetherian rings Λ such that the class of projective (respectively free) left Λ-modules is definable in L∞ω.

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