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.

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.

On n-Coherent Rings and (n,d)-Injective Modules

2015 ◽  
Vol 22 (02) ◽  
pp. 349-360
Author(s):  
Dongdong Zhang ◽  
Baiyu Ouyang
Keyword(s):  
Injective Modules ◽  
Coherent Ring ◽  
Derived Functors ◽  
Coherent Rings

Let R be a ring, n, d be fixed non-negative integers, [Formula: see text] the class of (n,d)-injective left R-modules, and [Formula: see text] the class of (n,d)-flat right R-modules. In this paper, we prove that if R is a left n-coherent ring and m ≥ 2, then [Formula: see text] if and only if [Formula: see text], if and only if Ext m+k(M,N) = 0 for all left R-modules M, N and all k ≥ -1, if and only if Ext m-1(M,N) = 0 for all left R-modules M, N. Meanwhile, we prove that if R is a left n-coherent ring, then − ⊗ − is right balanced on [Formula: see text] by [Formula: see text], and investigate the global right [Formula: see text]-dimension of [Formula: see text] and the global right [Formula: see text]-dimension of [Formula: see text] by right derived functors of − ⊗ −. Some known results are obtained as corollaries.

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.

Triangulated Equivalences Involving Gorenstein Projective Modules

2017 ◽  
Vol 60 (4) ◽  
pp. 879-890 ◽  
Author(s):  
Yuefei Zheng ◽  
Zhaoyong Huang
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Projective Modules ◽  
Injective Dimension ◽  
Stable Category ◽  
Dualizing Complex ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Left And Right

AbstractFor any ring R, we show that, in the bounded derived category Db(Mod R) of left R-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category (resp. ) of Gorenstein projective (resp. injective) modules. As a consequence, we get that if R is a left and right noetherian ring admitting a dualizing complex, then and are equivalent.

Cotorsion dimensions relative to semidualizing modules

2016 ◽  
Vol 15 (06) ◽  
pp. 1650104
Author(s):  
Xiuli Chen ◽  
Jianlong Chen
Keyword(s):  
Commutative Ring ◽  
Bass Class ◽  
Semidualizing Modules ◽  
Definition Of ◽  
Transfer Properties ◽  
Auslander Class

Let [Formula: see text] be a semidualizing [Formula: see text]-module, where [Formula: see text] is a commutative ring. We first introduce the definition of [Formula: see text]-cotorsion modules, and obtain the properties of [Formula: see text]-cotorsion modules. As applications, we give some new characterizations for perfect rings. Second, we study the Foxby equivalences between the subclasses of the Auslander class and that of the Bass class with respect to [Formula: see text]. Finally, we discuss [Formula: see text]-cotorsion dimensions and investigate the transfer properties of strongly [Formula: see text]-cotorsion dimensions under almost excellent extensions.

scholarly journals Localizations of injective modules

1985 ◽  
Vol 28 (3) ◽  
pp. 289-299 ◽  
Author(s):  
K. R. Goodearl ◽  
D. A. Jordan
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Simple Module ◽  
Global Dimension ◽  
Injective Hull ◽  
Injective Dimension ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Noetherian Domain ◽  
Left And Right

The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.

scholarly journals Exactness, Tor and Flat Modules Over a Commutative Ring

2004 ◽  
Vol 3 (3) ◽  
Author(s):  
Abhishek Banerjee
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Necessary Condition ◽  
Finitely Generated ◽  
Injective Modules ◽  
Flat Modules ◽  
Special Cases ◽  
Derived Functors ◽  
Finitely Generated Modules

In this paper, we principally explore flat modules over a commutative ring with identity. We do this in relation to projective and injective modules with the help of derived functors like Tor and Ext. We also consider an extension of the property of flatness and induce analogies with the “special cases” occurring in flat modules. We obtain some results on flatness in the context of a noetherian ring. We also characterize flat modules generated by one element and obtain a necessary condition for flatness of finitely generated modules.

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