Hereditary and Semihereditary Rings. Piecewise Domains

2016 ◽  
pp. 308-347
Keyword(s):  
Semihereditary Rings

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.

scholarly journals Torsion theories over semihereditary rings

1986 ◽  
Vol 40 (1) ◽  
pp. 54-70 ◽  
Author(s):  
M. W. Evans
Keyword(s):  
Ring Homomorphism ◽  
Torsion Theory ◽  
Dedekind Domain ◽  
Reflective Subcategory ◽  
Flat Modules ◽  
Torsion Theories ◽  
The Right ◽  
Torsion Free ◽  
Regular Rings ◽  
Semihereditary Rings

AbstractIn this paper the class of rings for which the right flat modules form the torsion-free class of a hereditary torsion theory (G, ℱ) are characterized and their structure investigated. These rings are called extended semihereditary rings. It is shown that the class of regular rings with ring homomorphism is a full co-reflective subcategory of the class of extended semihereditary rings with “flat” homomorphisms. A class of prime torsion theories is introduced which determines the torsion theory (G, ℱG). The torsion theory (JG, ℱG) is used to find a suitable generalisation of Dedekind Domain.

REGULARITY AND STRONG REGULARITY IN THE CONTEXT OF CERTAIN CLASSES OF RINGS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250205 ◽  
Author(s):  
MICHAŁ ZIEMBOWSKI
Keyword(s):  
Positive Integer ◽  
Sufficient Condition ◽  
Strong Regularity ◽  
Necessary And Sufficient Condition ◽  
Weak Dimension ◽  
Ring Of Polynomials ◽  
Necessary And Sufficient ◽  
Semihereditary Rings ◽  
The Ideal

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

Strongly D2 modules and their applications

2021 ◽  
pp. 2250071
Author(s):  
Mingzhao Chen ◽  
Hwankoo Kim ◽  
Fanggui Wang
Keyword(s):  
Positive Integer ◽  
Principal Ideal ◽  
Projective Module ◽  
Negative Answer ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Ideal Domain ◽  
Finitely Generated Modules ◽  
Structural Characterizations ◽  
Semihereditary Rings

An [Formula: see text]-module [Formula: see text] is called strongly [Formula: see text] if [Formula: see text] is a [Formula: see text] (equivalently, direct projective) module for every positive integer [Formula: see text]. In this paper, we consider the class of quasi-projective [Formula: see text]-modules, the class of strongly [Formula: see text] [Formula: see text]-modules and the class of [Formula: see text]-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li–Chen–Kourki. We also give structural characterizations of strongly [Formula: see text] modules for finitely generated modules over a principal ideal domain. In addition, we characterize some rings such as Artinian semisimple rings, hereditary rings, semihereditary rings and perfect rings in terms of strongly [Formula: see text] modules.

scholarly journals Semihereditary rings

1967 ◽  
Vol 73 (5) ◽  
pp. 656-659 ◽  
Author(s):  
Lance W. Small
Keyword(s):  
Semihereditary Rings

On semihereditary rings

1988 ◽  
Vol 16 (6) ◽  
pp. 1243-1274 ◽  
Author(s):  
Warren Dicks ◽  
A.H. Schofield
Keyword(s):  
Semihereditary Rings

A Note on Semihereditary Rings

1973 ◽  
Vol 16 (3) ◽  
pp. 439-440
Author(s):  
E. Enochs
Keyword(s):  
Commutative Ring ◽  
Noncommutative Version ◽  
Semihereditary Rings

It's well known (see Endo [1]) that for a commutative ring A, if A is semihereditary then w.gl. dim. A ≤ 1. It seems worth recording the noncommutative version of this.

Sign in / Sign up

Export Citation Format

Share Document