scholarly journals New characterizations of S-coherent rings

2021 ◽  
Author(s):  
Wei Qi ◽  
Xiaolei Zhang ◽  
Wei Zhao
Keyword(s):  
Direct Product ◽  
Text Coherence ◽  
Flat Modules ◽  
Text Version ◽  
Coherent Rings ◽  
Open Question

In this paper, we introduce and study the class [Formula: see text]-[Formula: see text]-ML of [Formula: see text]-Mittag-Leffler modules with respect to all flat modules. We show that a ring [Formula: see text] is [Formula: see text]-coherent if and only if every ideal is in [Formula: see text]-[Formula: see text]-ML, if and only if [Formula: see text]-[Formula: see text]-ML is closed under submodules. As an application, we obtain the [Formula: see text]-version of Chase Theorem: a ring [Formula: see text] is [Formula: see text]-coherent if and only if any direct product of copies of [Formula: see text] is [Formula: see text]-flat, if and only if any direct product of flat [Formula: see text]-modules is [Formula: see text]-flat. Consequently, we provide an answer to the open question proposed by Bennis and El Hajoui [On [Formula: see text]-coherence, J. Korean Math. Soc. 55(6) (2018) 1499–1512].

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.

Some Remarks on Groups Admitting a Fixed-Point-Free Automorphism

1968 ◽  
Vol 20 ◽  
pp. 1300-1307 ◽  
Author(s):  
Fletcher Gross
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Simple Group ◽  
Identity Element ◽  
Simple Groups ◽  
Cyclic Groups ◽  
A Finite Group ◽  
Open Question

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.

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.

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.

Two applications of Nagata rings and modules

2019 ◽  
Vol 19 (06) ◽  
pp. 2050115
Author(s):  
Fanggui Wang ◽  
Lei Qiao
Keyword(s):  
New York ◽  
Commutative Ring ◽  
Applied Mathematics ◽  
Torsion Theory ◽  
Ideal Theory ◽  
Star Operation ◽  
Text Version ◽  
Commutative Theory ◽  
Coherent Rings ◽  
Relative Invariants

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

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.

GORENSTEIN FP-INJECTIVE AND GORENSTEIN FLAT MODULES

2008 ◽  
Vol 07 (04) ◽  
pp. 491-506 ◽  
Author(s):  
LIXIN MAO ◽  
NANQING DING
Keyword(s):  
Exact Sequence ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Rings ◽  
Gorenstein Flat

In this paper, Gorenstein FP-injective modules are introduced and studied. An R-module M is called Gorenstein FP-injective if there is an exact sequence ⋯ → E1 → E0 → E0 → E1 → ⋯ of injective R-modules with M = ker (E0 → E1) and such that Hom (E, -) leaves the sequence exact whenever E is an FP-injective R-module. Some properties of Gorenstein FP-injective and Gorenstein flat modules over coherent rings are obtained. Several known results are extended.

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.

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