Relatively free modules on ring extensions

2021 ◽  
pp. 1-14
Author(s):  
Shufeng Guo ◽  
Xiaochen Wang ◽  
Zhong Yi
Keyword(s):  
Ring Extensions ◽  
Free Modules

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

scholarly journals Injective modules under faithfully flat ring extensions

2015 ◽  
Vol 144 (3) ◽  
pp. 1015-1020 ◽  
Author(s):  
Lars Winther Christensen ◽  
Fatih Köksal
Keyword(s):  
Injective Modules ◽  
Flat Ring ◽  
Ring Extensions

scholarly journals Almost free modules over modular group algebras

1976 ◽  
Vol 41 (2) ◽  
pp. 243-254 ◽  
Author(s):  
Jon F Carlson
Keyword(s):  
Modular Group ◽  
Group Algebras ◽  
Free Modules

On Radicals of Skew Inverse Laurent Series Rings

2016 ◽  
Vol 23 (02) ◽  
pp. 335-346
Author(s):  
A. Moussavi
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Laurent Series ◽  
Ring Extensions ◽  
Series Type ◽  
Armendariz Ring

Let R be a ring and α an automorphism of R. Amitsur proved that the Jacobson radical J(R[x]) of the polynomial ring R[x] is the polynomial ring over the nil ideal J(R[x]) ∩ R. Following Amitsur, it is shown that when R is an Armendariz ring of skew inverse Laurent series type and S is any one of the ring extensions R[x;α], R[x,x-1;α], R[[x-1;α]] and R((x-1;α)), then ℜ𝔞𝔡(S) = ℜ𝔞𝔡(R)S = Nil (S), ℜ𝔞𝔡(S) ∩ R = Nil (R), where ℜ𝔞𝔡 is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals.

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 Square-free modules with the exchange property

2010 ◽  
Vol 323 (7) ◽  
pp. 1993-2001 ◽  
Author(s):  
Pace P. Nielsen
Keyword(s):  
Exchange Property ◽  
Free Modules

Free Summands of Stably Free Modules

1982 ◽  
Vol 25 (3) ◽  
pp. 361-362
Author(s):  
Robert W. Swift
Keyword(s):  
Free Modules

AbstractIn this note we show that the generic orthogonal stably free modules of type (2, 7) and (3, 8) have one free summand. This completes the work of other authors on free summands of orthogonal stably free modules.

On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions

2015 ◽  
Vol 58 (1) ◽  
pp. 134-143
Author(s):  
Saeed Nasseh
Keyword(s):  
Local Ring ◽  
Gorenstein Ring ◽  
Ring Extensions

AbstractWe show that under some conditions a Gorenstein ring R satisfies the Generalized Auslander–Reiten conjecture if and only if R[x] does. When R is a local ring we prove the same result for some localizations of R[x].

Sign in / Sign up

Export Citation Format

Share Document