FINITISTIC n-SELF-COTILTING MODULES OVER RING EXTENSIONS

Salah Al-Nofayee

doi:10.17654/nt038050499

Extensions of Rings Having McCoy Condition

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-029-5 ◽

2009 ◽

Vol 52 (2) ◽

pp. 267-272 ◽

Cited By ~ 5

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.

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Injective modules under faithfully flat ring extensions

Proceedings of the American Mathematical Society ◽

10.1090/proc/12791 ◽

2015 ◽

Vol 144 (3) ◽

pp. 1015-1020 ◽

Cited By ~ 3

Author(s):

Lars Winther Christensen ◽

Fatih Köksal

Keyword(s):

Injective Modules ◽

Flat Ring ◽

Ring Extensions

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On Radicals of Skew Inverse Laurent Series Rings

Algebra Colloquium ◽

10.1142/s1005386716000353 ◽

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.

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PROPERLY STRATIFIED ALGEBRAS AND TILTING

Proceedings of the London Mathematical Society ◽

10.1017/s0024611505015431 ◽

2005 ◽

Vol 92 (1) ◽

pp. 29-61 ◽

Cited By ~ 8

Author(s):

ANDERS FRISK ◽

VOLODYMYR MAZORCHUK

Keyword(s):

Projective Dimension ◽

Finitistic Dimension ◽

Tilting Module ◽

Tilting Modules ◽

Category Of Modules ◽

Cotilting Modules ◽

Contravariantly Finite ◽

Properly Stratified Algebras ◽

Stratified Algebra ◽

Generalized Tilting Module

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

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Commutative rings and modules that are Nil-coherent or special Nil-coherent

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501870 ◽

2017 ◽

Vol 16 (10) ◽

pp. 1750187 ◽

Cited By ~ 2

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.

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On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-052-9 ◽

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].

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