scholarly journals I-n-Coherent Rings, I-n-Semihereditary Rings, and I-Regular Rings

2014 ◽  
Vol 66 (6) ◽  
pp. 857-883
Author(s):  
Zhu Zhanmin
Keyword(s):  
Coherent Rings ◽  
Regular Rings ◽  
Semihereditary Rings

C-coherent rings, C-semihereditary rings and C-regular rings

2013 ◽  
Vol 50 (4) ◽  
pp. 491-508 ◽  
Author(s):  
Zhanmin Zhu
Keyword(s):  
Direct Summand ◽  
Finitely Generated ◽  
Coherent Rings ◽  
Finitely Presented ◽  
Regular Rings ◽  
Semihereditary Rings

Let C be a class of some finitely presented left R-modules. A left R-module M is called C-injective, if ExtR1(C, M) = 0 for each C ∈ C. A right R-module M is called C-flat, if Tor1R(M, C) = 0 for each C ∈ C. A ring R is called C-coherent, if every C ∈ C is 2-presented. A ring R is called C-semihereditary, if whenever 0 → K → P → C → 0 is exact, where C ∈ C and P is finitely generated projective and K is finitely generated, then K is also projective. A ring R is called C-regular, if whenever P/K ∈ C, where P is finitely generated projective and K is finitely generated, then K is a direct summand of P. Using the concepts of C-injectivity and C-flatness of modules, we present some characterizations of C-coherent rings, C-semihereditary rings, and C-regular 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.

scholarly journals Rd-phantom and RD-Ext-phantom morphisms

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2883-2895
Author(s):  
Lixin Mao
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Coherent Rings ◽  
Regular Rings

A morphism f of left R-modules is called an RD-phantom morphism if the induced morphism Tor1(R=aR,f)=0 for any a ? R. Similarly, a morphism g of left R-modules is said to be an RD-Ext-phantom morphism if the induced morphism Ext1(R=Ra,g)=0 for any a ? R. It is proven that a morphism f is an RD-phantom morphism if and only if the pullback of any short exact sequence along f is an RD-exact sequence; a morphism g is an RD-Ext-phantom morphism if and only if the pushout of any short exact sequence along g is an RD-exact sequence. We also characterize Pr?fer domains, left P-coherent rings, left PP rings, von Neumann regular rings in terms of RD-phantom and RD-Ext-phantom morphisms. Finally, we prove that every module has an epic RD-phantom cover with the kernel RD-injective and has a monic RD-Ext-phantom preenvelope with the cokernel RD-projective.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

On (Strongly) Gorenstein Von Neumann Regular Rings

2011 ◽  
Vol 39 (9) ◽  
pp. 3242-3252 ◽  
Author(s):  
Najib Mahdou ◽  
Mohammed Tamekkante ◽  
Siamak Yassemi
Keyword(s):  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

On feckly polar rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950021
Author(s):  
Tugce Pekacar Calci ◽  
Huanyin Chen
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Structure Theorems ◽  
Regular Rings

In this paper, we introduce a new notion which lies properly between strong [Formula: see text]-regularity and pseudopolarity. A ring [Formula: see text] is feckly polar if for any [Formula: see text] there exists [Formula: see text] such that [Formula: see text] Many structure theorems are proved. Further, we investigate feck polarity for triangular matrix and matrix rings. The relations among strongly [Formula: see text]-regular rings, pseudopolar rings and feckly polar rings are also obtained.

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