scholarly journals Torsion theories and coherent rings

1973 ◽  
Vol 8 (2) ◽  
pp. 233-239 ◽  
Author(s):  
J.M. Campbell
Keyword(s):  
Direct Product ◽  
Torsion Theory ◽  
Finitely Generated ◽  
Coherent Ring ◽  
Ring Of Quotients ◽  
Torsion Theories ◽  
Coherent Rings ◽  
Finitely Related

Chase has given several characterizations of a right coherent ring, among which are: every direct product of copies of the ring is left-flat; and every finitely generated submodule of a free right module is finitely related. We extend his results to obtain conditions for the ring of quotients of a ring with respect to a torsion theory to be coherent.

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.

scholarly journals Lattice-ordered modules of quotients

1980 ◽  
Vol 30 (2) ◽  
pp. 243-251 ◽  
Author(s):  
Stuart A. Steinberg
Keyword(s):  
Jacobson Radical ◽  
Torsion Theory ◽  
Finitely Generated ◽  
Hereditary Torsion Theory ◽  
Ring Of Quotients

AbstractLet Q be the ring of quotients of the f-ring R with respect to a positive hereditary torsion theory and suppose Q is a right f-ring. It is shown that if the finitely-generated right ideals of R are principal, then Q is an f-ring. Also, if QR is injective, Q is an f-ring if and only if its Jacobson radical is convex. Moreover, a class of po-rings is introduced (which includes the classes of commutative po-rings and right convex f-rings) over which Q(M) is an f-module for each f-module M.

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].

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.

RELATIVE COHERENCE OF RINGS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250047
Author(s):  
LIXIN MAO ◽  
NANQING DING
Keyword(s):  
Left Ideal ◽  
Torsion Theory ◽  
Hereditary Torsion Theory ◽  
Injective Modules ◽  
Coherent Ring ◽  
Coherent Rings ◽  
Finitely Presented

Let R be a ring and τ a hereditary torsion theory for the category of all left R-modules. A right R-module M is called τ-flat if Tor 1(M, R/I) = 0 for any τ-finitely presented left ideal I. A left R-module N is said to be τ-f-injective in case Ext 1(R/I, N) = 0 for any τ-finitely presented left ideal I. R is called a left τ-coherent ring in case every τ-finitely presented left ideal is finitely presented. τ-coherent rings are characterized in terms of, among others, τ-flat and τ-f-injective modules. Some known results are extended.

A finitely generated, infinitely related group with trivial multiplicator

1971 ◽  
Vol 5 (1) ◽  
pp. 131-136 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Metabelian Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Related Group ◽  
Finitely Related

We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator.1. The purpose of this note is to establish the exitense of a finitely generated group which is not finitely related, but whose multiplecator is finitely generated. This settles negatively a question whichb has been open for a few years (it was first brought to my attention by Michel Kervaire and Joan Landman Dyer in 1964, but I believe it is somewhat older). The group is given in the follwing theorem.

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.

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.

FREE SEMIGROUP PRESENTATIONS

2018 ◽  
Vol 1 (1) ◽  
Author(s):  
Nwawuru Francis
Keyword(s):  
Direct Product ◽  
Free Semigroup ◽  
Finitely Generated ◽  
Free Semigroups ◽  
Finitely Presented

Let and  be two free semigroups. We define external direct product of two free semigroups as an ordered pair of words such  that and .We investigate the presentations of external direct product of free semigroups, state and prove under some conditions that the external direct product of two finitely generated free semigroups is finitely generated, also the external direct product of two finitely presented free semigroups is finitely presented. 

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