scholarly journals Cartan-Eilenberg Gorenstein Flat Complexes

2014 ◽  
Vol 114 (1) ◽  
pp. 5 ◽  
Author(s):  
Gang Yang ◽  
Li Liang
Keyword(s):  
Coherent Ring ◽  
Coherent Rings ◽  
Flat Cover ◽  
Gorenstein Flat

In this paper, we study Cartan-Eilenberg Gorenstein flat complexes. We show that over coherent rings a Cartan-Eilenberg Gorenstein flat complex can be gotten by a so-called complete Cartan-Eilenberg flat resolution. We argue that over a coherent ring every complex has a Cartan-Eilenberg Gorenstein flat cover.

scholarly journals SOME COVERS AND ENVELOPES IN THE CHAIN COMPLEX CATEGORY OF R-MODULES

2011 ◽  
Vol 90 (3) ◽  
pp. 385-401
Author(s):  
ZHANPING WANG ◽  
ZHONGKUI LIU
Keyword(s):  
Chain Complex ◽  
Cotorsion Pair ◽  
Coherent Ring ◽  
Direct Limits ◽  
Pure Submodules ◽  
Flat Cover ◽  
Gorenstein Flat

AbstractWe study the existence of some covers and envelopes in the chain complex category of R-modules. Let (𝒜,ℬ) be a cotorsion pair in R-Mod and let ℰ𝒜 stand for the class of all exact complexes with each term in 𝒜. We prove that (ℰ𝒜,ℰ𝒜⊥) is a perfect cotorsion pair whenever 𝒜 is closed under pure submodules, cokernels of pure monomorphisms and direct limits and so every complex has an ℰ𝒜-cover. As an application we show that every complex of R-modules over a right coherent ring R has an exact Gorenstein flat cover. In addition, the existence of $\overline {\mathcal {A}}$-covers and $\overline {\mathcal {B}}$-envelopes of special complexes is considered where $\overline {\mathcal {A}}$ and $\overline {\mathcal {B}}$ denote the classes of all complexes with each term in 𝒜 and ℬ, respectively.

Commutative rings and modules that are Nil*-coherent or special Nil*-coherent

2017 ◽  
Vol 16 (10) ◽  
pp. 1750187 ◽  
Author(s):  
Karima Alaoui Ismaili ◽  
David E. Dobbs ◽  
Najib Mahdou
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Basic Properties ◽  
Unital Ring ◽  
Reduced Ring ◽  
Coherent Ring ◽  
Ring Extensions ◽  
Coherent Rings ◽  
Finitely Presented

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

Relative FP-gr-injective and gr-flat modules

2018 ◽  
Vol 28 (06) ◽  
pp. 959-977 ◽  
Author(s):  
Tiwei Zhao ◽  
Zenghui Gao ◽  
Zhaoyong Huang
Keyword(s):  
Graded Modules ◽  
Flat Modules ◽  
Duality Pairs ◽  
Coherent Rings ◽  
Flat Cover ◽  
Finitely Presented

Let [Formula: see text] be an integer. We introduce the notions of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Then we investigate the properties of these modules by using the properties of special finitely presented graded modules and obtain some equivalent characterizations of [Formula: see text]-gr-coherent rings in terms of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Moreover, we prove that the pairs (gr-[Formula: see text], gr-[Formula: see text]) and (gr-[Formula: see text], gr-[Formula: see text]) are duality pairs over left [Formula: see text]-coherent rings, where gr-[Formula: see text] and gr-[Formula: see text] denote the subcategories of [Formula: see text]-FP-gr-injective left [Formula: see text]-modules and [Formula: see text]-gr-flat right [Formula: see text]-modules respectively. As applications, we obtain that any graded left (respectively, right) [Formula: see text]-module admits an [Formula: see text]-FP-gr-injective (respectively, [Formula: see text]-gr-flat) cover and preenvelope.

A non-commutative analogue of Costa’s first conjecture

2019 ◽  
Vol 19 (01) ◽  
pp. 2050007
Author(s):  
Weiqing Li ◽  
Dong Liu
Keyword(s):  
Negative Integer ◽  
Noetherian Rings ◽  
Coherent Ring ◽  
Coherent Rings

Let [Formula: see text] and [Formula: see text] be arbitrary fixed integers. We prove that there exists a ring [Formula: see text] such that: (1) [Formula: see text] is a right [Formula: see text]-ring; (2) [Formula: see text] is not a right [Formula: see text]-ring for each non-negative integer [Formula: see text]; (3) [Formula: see text] is not a right [Formula: see text]-ring [Formula: see text]for [Formula: see text], for each non-negative integer [Formula: see text]; (4) [Formula: see text] is a right [Formula: see text]-coherent ring; (5) [Formula: see text] is not a right [Formula: see text]-coherent ring. This shows the richness of right [Formula: see text]-rings and right [Formula: see text]-coherent rings, and, in particular, answers affirmatively a problem posed by Costa in [D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra 22 (1994) 3997–4011.] when the ring in question is non-commutative.

Coherent rings and Gorenstein flat character modules

2020 ◽  
pp. 1-10
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan ◽  
Guocheng Dai
Keyword(s):  
Coherent Rings ◽  
Gorenstein Flat

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.

scholarly journals Global dimensions of right coherent rings with left Krull dimension

1989 ◽  
Vol 39 (2) ◽  
pp. 215-223 ◽  
Author(s):  
Mark L. Teply
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Homological Dimension ◽  
Isomorphism Classes ◽  
Coherent Ring ◽  
Gabriel Dimension ◽  
Coherent Rings

The weak global dimension of a right coherent ring with left Krull dimension α ≥ 1 is found to be the supremum of the weak dimensions of the β-critical cyclic modules, where β < α. If, in addition, the mapping I → assl gives a bijection between isomorphism classes on injective left R-modules and prime ideals of R, then the weak global dimension of R is the supremum of the weak dimensions of the simple left R-modules. These results are used to compute the left homological dimension of a right coherent, left noetherian ring. Some analogues of our results are also given for rings with Gabriel dimension.

scholarly journals The existence of Gorenstein flat covers

2004 ◽  
Vol 94 (1) ◽  
pp. 46 ◽  
Author(s):  
Edgar E. Enochs ◽  
Overtoun M. G. Jenda ◽  
J. A. Lopez-Ramos
Keyword(s):  
Coherent Ring ◽  
Gorenstein Flat

We prove that all left modules over a right coherent ring have Gorenstein flat covers.

scholarly journals Acyclic Complexes and Gorenstein Rings

2020 ◽  
Vol 27 (03) ◽  
pp. 575-586
Author(s):  
Sergio Estrada ◽  
Alina Iacob ◽  
Holly Zolt
Keyword(s):  
Analogous Result ◽  
Gorenstein Ring ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Ring ◽  
Acyclic Complexes ◽  
Gorenstein Flat ◽  
Gorenstein Injective

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

K-theory of n-coherent rings

2021 ◽  
pp. 2350007
Author(s):  
Eugenia Ellis ◽  
Rafael Parra
Keyword(s):  
Full Subcategory ◽  
Projective Dimension ◽  
Exact Category ◽  
Finite Projective Dimension ◽  
Coherent Ring ◽  
K Theory ◽  
Coherent Rings

Let [Formula: see text] be a strong [Formula: see text]-coherent ring such that each finitely [Formula: see text]-presented [Formula: see text]-module has finite projective dimension. We consider [Formula: see text] the full subcategory of [Formula: see text]-Mod of finitely [Formula: see text]-presented modules. We prove that [Formula: see text] is an exact category, [Formula: see text] for every [Formula: see text] and we obtain an expression of [Formula: see text].

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