scholarly journals Gorenstein n-X -Injective and n - X-Flat Modules with Respect to a Special Finitely Presented Module

2021 ◽  
Vol 28 (04) ◽  
pp. 673-688
Author(s):  
Mostafa Amini ◽  
Arij Benkhadra ◽  
Driss Bennis
Keyword(s):  
Flat Modules ◽  
Coherent Rings ◽  
Equivalent Properties ◽  
Finitely Presented

Let [Formula: see text] be a ring, [Formula: see text] a class of [Formula: see text]-modules and [Formula: see text] an integer. We introduce the concepts of Gorenstein [Formula: see text]-[Formula: see text]-injective and [Formula: see text]-[Formula: see text]-flat modules via special finitely presented modules. Besides, we obtain some equivalent properties of these modules on [Formula: see text]-[Formula: see text]-coherent rings. Then we investigate the relations among Gorenstein [Formula: see text]-[Formula: see text]-injective, [Formula: see text]-[Formula: see text]-flat, injective and flat modules on [Formula: see text]-[Formula: see text]-rings (i.e., self [Formula: see text]-[Formula: see text]-injective and [Formula: see text]-[Formula: see text]-coherent rings). Several known results are generalized to this new context.

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.

J-COHERENT RINGS

2009 ◽  
Vol 08 (02) ◽  
pp. 139-155 ◽  
Author(s):  
NANQING DING ◽  
YUANLIN LI ◽  
LIXIN MAO
Keyword(s):  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Flat Modules ◽  
Coherent Ring ◽  
Necessary And Sufficient ◽  
Coherent Rings ◽  
Finitely Presented

Let R be a ring. Recall that a left R-module M is coherent if every finitely generated submodule of M is finitely presented. R is a left coherent ring if the left R-module RR is coherent. In this paper, we say that R is left J-coherent if its Jacobson radical J(R) is a coherent left R-module. J-injective and J-flat modules are introduced to investigate J-coherent rings. Necessary and sufficient conditions for R to be left J-coherent are given. It is shown that there are many similarities between coherent and J-coherent rings. J-injective and J-flat dimensions are also studied.

Noncommutative G-semihereditary rings

2018 ◽  
Vol 17 (01) ◽  
pp. 1850014 ◽  
Author(s):  
Jian Wang ◽  
Yunxia Li ◽  
Jiangsheng Hu
Keyword(s):  
Associative Ring ◽  
Sufficient Condition ◽  
Projective Modules ◽  
The Third ◽  
Flat Modules ◽  
Gorenstein Projective Modules ◽  
Direct Limits ◽  
Gorenstein Flat ◽  
Finitely Presented ◽  
Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

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.

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.

Homological Properties Relative to Injectively Resolving Subcategories

2015 ◽  
Vol 58 (4) ◽  
pp. 741-756 ◽  
Author(s):  
Zenghui Gao
Keyword(s):  
Von Neumann ◽  
Direct Sum ◽  
Von Neumann Regular Rings ◽  
Flat Modules ◽  
Resolving Subcategory ◽  
Von Neumann Regular ◽  
Finitely Presented ◽  
Regular Rings

AbstractLet ε be an injectively resolving subcategory of left R-modules. A left R-module M (resp. right R-module N) is called ε-injective (resp. ε-flat) if Ext1R (G,M) = 0 (resp. TorR1 (N, G) = 0) for any G ∊ ε. Let ε be a covering subcategory. We prove that a left R-module M is E-injective if and only if M is a direct sum of an injective left R-module and a reduced E-injective left R-module. Suppose ℱ is a preenveloping subcategory of right R-modules such that ε+ ⊆ ℱ and ℱ+ ⊆ ε. It is shown that a finitely presented right R-module M is ε-flat if and only if M is a cokernel of an ℱ-preenvelope of a right R-module. In addition, we introduce and investigate the ε-injective and ε-flat dimensions of modules and rings. We also introduce ε-(semi)hereditary rings and ε-von Neumann regular rings and characterize them in terms of ε-injective and ε-flat modules.

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

2005 ◽  
Vol 12 (01) ◽  
pp. 149-160 ◽  
Author(s):  
Xiaoxiang Zhang ◽  
Jianlong Chen ◽  
Juan Zhang
Keyword(s):  
Injective Modules ◽  
Coherent Rings ◽  
Positive Integers ◽  
Equivalent Conditions ◽  
Finitely Presented

Let R be a ring. For two fixed positive integers m and n, a right R-module M is called (m,n)-injective in case every right R-homomorphism from an n-generated submodule of Rm to M extends to one from Rm to M. R is said to be left (m,n)-coherent if each n-generated submodule of the left R-module Rm is finitely presented. In this paper, we give some new characterizations of (m,n)-injective modules. We also derive various equivalent conditions for a ring to be left (m,n)-coherent. Some known results on coherent rings are obtained as corollaries.

scholarly journals STRONGLY GORENSTEIN FLAT MODULES

2009 ◽  
Vol 86 (3) ◽  
pp. 323-338 ◽  
Author(s):  
NANQING DING ◽  
YUANLIN LI ◽  
LIXIN MAO
Keyword(s):  
Exact Sequence ◽  
Projective Modules ◽  
Flat Dimension ◽  
Flat Modules ◽  
Coherent Rings ◽  
Gorenstein Flat ◽  
Strongly Gorenstein Flat Modules

AbstractIn this paper, strongly Gorenstein flat modules are introduced and investigated. An R-module M is called strongly Gorenstein flat if there is an exact sequence ⋯→P1→P0→P0→P1→⋯ of projective R-modules with M=ker (P0→P1) such that Hom(−,F) leaves the sequence exact whenever F is a flat R-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied.

scholarly journals Some results on coherent rings II

1967 ◽  
Vol 8 (2) ◽  
pp. 123-126 ◽  
Author(s):  
Morton E. Harris
Keyword(s):  
Finitely Generated ◽  
Basic Theorem ◽  
Coherent Ring ◽  
Coherent Rings ◽  
Finitely Presented

According to Bourbaki [1, pp. 62–63, Exercise 11], a left (resp. right) A-module M is said to be pseudo-coherent if every finitely generated submodule of M is finitely presented, and is said to be coherent if it is both pseudo-coherent and finitely generated. This Bourbaki reference contains various results on pseudo-coherent and coherent modules. Then, in [1, p. 63, Exercise 12], a ring which as a left (resp. right) module over itself is coherent is said to be a left (resp. right) coherent ring, and various results on and examples of coherent rings are presented. The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3].

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