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].

Krull Dimension of Injective Modules Over Commutative Noetherian Rings

2005 ◽  
Vol 48 (2) ◽  
pp. 275-282
Author(s):  
Patrick F. Smith
Keyword(s):  
Noetherian Ring ◽  
Integral Domain ◽  
Krull Dimension ◽  
Injective Module ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
One Dimensional ◽  
Injective Modules

AbstractLet R be a commutative Noetherian integral domain with field of fractions Q. Generalizing a forty-year-old theorem of E. Matlis, we prove that the R-module Q/R (or Q) has Krull dimension if and only if R is semilocal and one-dimensional. Moreover, if X is an injective module over a commutative Noetherian ring such that X has Krull dimension, then the Krull dimension of X is at most 1.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

WEAK AB-CONTEXT FOR FP-INJECTIVE MODULES WITH RESPECT TO SEMIDUALIZING MODULES

2013 ◽  
Vol 12 (07) ◽  
pp. 1350039 ◽  
Author(s):  
JIANGSHENG HU ◽  
DONGDONG ZHANG
Keyword(s):  
Noetherian Ring ◽  
Model Structure ◽  
Injective Dimension ◽  
Commutative Noetherian Ring ◽  
New Model ◽  
Injective Modules ◽  
Semidualizing Modules ◽  
Semidualizing Bimodule ◽  
Gorenstein Injective

Let S and R be rings and SCR a semidualizing bimodule. We define and study GC-FP-injective R-modules, and these modules are exactly C-Gorenstein injective R-modules defined by Holm and Jørgensen provided that S = R is a commutative Noetherian ring. We mainly prove that the category of GC-FP-injective R-modules is a part of a weak AB-context, which is dual of weak AB-context in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander–Buchweitz approximations for R-modules with finite GC-FP-injective dimension. As an application, a new model structure in Mod R is given.

scholarly journals The New Results in n-injective Modules and n-projective Modules

2021 ◽  
pp. 271-285
Author(s):  
Samira Hashemi ◽  
Feysal Hassani ◽  
Rasul Rasuli
Keyword(s):  
Exact Sequence ◽  
Projective Module ◽  
Injective Module ◽  
Projective Modules ◽  
Injective Modules

In this paper, we introduce and clarify a new presentation between the n-exact sequence and the n-injective module and n-projective module. Also, we obtain some new results about them.

scholarly journals On injectiveL-modules

2005 ◽  
Vol 2005 (5) ◽  
pp. 747-754 ◽  
Author(s):  
Paul Isaac
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Injective Module ◽  
Projective Modules ◽  
Module Theory ◽  
Direct Sum ◽  
Injective Modules ◽  
Free Modules

The concepts of free modules, projective modules, injective modules, and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of projective and injectiveL-modules. In this paper, we give an alternate definition for injectiveL-modules and prove that a direct sum ofL-modules is injective if and only if eachL-module in the sum is injective. Also we prove that ifJis an injective module andμis an injectiveL-submodule ofJ, and if0→μ→fv→gη→0is a short exact sequence ofL-modules, thenν≃μ⊕η.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings II

1992 ◽  
Vol 35 (3) ◽  
pp. 511-518
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Integral Closure ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

Let E be an injective module over a commutative Noetherian ring A (with non-zero identity), and let a be an ideal of A. The submodule (0:Eα) of E has a secondary representation, and so we can form the finite set AttA(0:Eα) of its attached prime ideals. In [1, 3.1], we showed that the sequence of sets is ultimately constant; in [2], we introduced the integral closure a*(E) of α relative to E, and showed that is increasing and ultimately constant. In this paper, we prove that, if a contains an element r such that rE = E, then is ultimately constant, and we obtain information about its ultimate constant value.

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.

Stability of Gorenstein Classes of Modules

2013 ◽  
Vol 20 (04) ◽  
pp. 623-636 ◽  
Author(s):  
Samir Bouchiba
Keyword(s):  
Exact Sequence ◽  
Projective Modules ◽  
Natural Question ◽  
Alternate Proof ◽  
Similar Treatment ◽  
Arbitrary Ring ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Category Of Modules ◽  
Gorenstein Injective

The purpose of this paper is to give, via totally different techniques, an alternate proof to the main theorem of [18] in the category of modules over an arbitrary ring R. In effect, we prove that this theorem follows from establishing a sequence of equalities between specific classes of R-modules. Actually, we tackle the following natural question: What notion emerges when iterating the very process applied to build the Gorenstein projective and Gorenstein injective modules from complete resolutions? In other words, given an exact sequence of Gorenstein injective R-modules G= ⋯ → G1→ G0→ G-1→ ⋯ such that the complex Hom R(H,G) is exact for each Gorenstein injective R-module H, is the module Im (G0→ G-1) Gorenstein injective? We settle such a question in the affirmative and the dual result for the Gorenstein projective modules follows easily via a similar treatment to that used in this paper. As an application, we provide the Gorenstein versions of the change of rings theorems for injective modules over an arbitrary ring.

scholarly journals Totally acyclic complexes and locally Gorenstein rings

2018 ◽  
Vol 17 (03) ◽  
pp. 1850039
Author(s):  
Lars Winther Christensen ◽  
Kiriko Kato
Keyword(s):  
Noetherian Ring ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Gorenstein Rings ◽  
Dualizing Complex ◽  
Injective Modules ◽  
Acyclic Complex ◽  
Acyclic Complexes

A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.

scholarly journals Faithfully Exact Functors and Their Applications to Projective Modules and Injective Modules

1964 ◽  
Vol 24 ◽  
pp. 29-42 ◽  
Author(s):  
Takeshi Ishikawa
Keyword(s):  
General Theory ◽  
Noetherian Ring ◽  
Projective Module ◽  
Special Kind ◽  
Projective Modules ◽  
Flat Module ◽  
Commutative Case ◽  
Injective Modules ◽  
Equivalent Conditions

The aim of this paper is to study a property of a special kind of exact functors and give some applications to projective modules and injective modules.In section 1 we introduce the notion of faithfully exact functors [Definition 1] as a generalization of the functor T(X) = X⊗M, where M is a faithfully flat module, and give a property of this class of functors [Theorem 1.1]. Next, applying this general theory to functors ⊗ and Horn, we define the notion of faithfully projective modules [Definition 2] and faithfully injective modules [Definition 3]. In the commutative case “faithfully projective” means, however, simply “projective and faithfully flat” [Proposition 2.3]. In section 2, equivalent conditions for a projective module P to be faithfully projective are given [Theorem 2.2, Proposition 2. 3 and 2.4]. And a simpler proof is given to Y. Hinohara’s result [6] asserting that projective modules over an indecomposable weakly noetherian ring are faithfully flat [Proposition 2.5]. In section 3, we consider faithfully injective modules.

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