Artin Rings with Two-Generated Ideals

1992 ◽  
Vol 35 (1) ◽  
pp. 133-135 ◽  
Author(s):  
David E. Rush
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Homomorphic Image ◽  
Endomorphism Ring ◽  
Commutative Group ◽  
Group Rings ◽  
One Dimensional ◽  
Intersection Ring

AbstractIt is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.

scholarly journals Complete Intersection in Overrings of a Certain One-Dimensional Gorenstein Graded Local Ring

2000 ◽  
Vol 233 (2) ◽  
pp. 772-790
Author(s):  
Shiro Goto ◽  
Satoshi Haraikawa ◽  
Shin-Ichiro Iai
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
One Dimensional

On *-clean non-commutative group rings

2016 ◽  
Vol 15 (08) ◽  
pp. 1650150 ◽  
Author(s):  
Hongdi Huang ◽  
Yuanlin Li ◽  
Gaohua Tang
Keyword(s):  
Finite Field ◽  
Local Ring ◽  
Group Algebra ◽  
Rational Numbers ◽  
Semisimple Group ◽  
Group Algebras ◽  
Group Rings ◽  
Dihedral Groups ◽  
Commutative Local Ring ◽  
Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

Commutative nil clean group rings

2015 ◽  
Vol 14 (06) ◽  
pp. 1550094 ◽  
Author(s):  
Warren Wm. McGovern ◽  
Shan Raja ◽  
Alden Sharp
Keyword(s):  
Commutative Group ◽  
University Of California ◽  
Short Paper ◽  
Group Rings ◽  
Clean Rings ◽  
Clean Ring ◽  
Nil Clean Ring

In [A. J. Diesl, Classes of strongly clean rings, Ph.D. Dissertation, University of California, Berkely (2006); Nil clean rings, J. Algebra383 (2013) 197–211], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short paper, we characterize nil clean commutative group rings.

Complete Intersection Flat Dimension and the Intersection Theorem

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1161-1166
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif ◽  
Siamak Yassemi
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Intersection Theorem ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Gorenstein Dimension ◽  
Flat Dimension ◽  
Gorenstein Flat ◽  
Similar Extension

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim RM and a Gorenstein dimension G-dim RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

On units in commutative group rings

1982 ◽  
Vol 38 (1) ◽  
pp. 420-422 ◽  
Author(s):  
G. Karpilovsky
Keyword(s):  
Commutative Group ◽  
Group Rings

scholarly journals Torsion in tensor powers of modules

2015 ◽  
Vol 219 ◽  
pp. 113-125
Author(s):  
Olgur Celikbas ◽  
Srikanth B. Iyengar ◽  
Greg Piepmeyer ◽  
Roger Wiegand
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Tensor Products ◽  
Central Theme ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Free Modules ◽  
Tensor Powers ◽  
Torsion Free ◽  
Nonzero Torsion

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modulesNwith the property thatM ⊗RNhas nonzero torsion unlessMis very special. An important example of such a moduleNis the Frobenius powerpeRover a complete intersection domainRof characteristicp> 0.

scholarly journals Group rings with finite central endomorphism dimension

1983 ◽  
Vol 24 (2) ◽  
pp. 169-176 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Endomorphism Ring ◽  
Finite Dimension ◽  
Group Rings

Let F be any field. Denote by the class of all groups G such that every irreducible FG-module has finite dimension over F and by the class of all groups G such that every irreducible FG-module has finite dimension over its endomorphism ring. Clearly

On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

2009 ◽  
Vol 37 (5) ◽  
pp. 1594-1603 ◽  
Author(s):  
M. D'Anna ◽  
M. Mezzasalma ◽  
V. Micale
Keyword(s):  
Local Ring ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
One Dimensional

The dilworth number of group rings over an artin local ring

2000 ◽  
Vol 28 (10) ◽  
pp. 4597-4609 ◽  
Author(s):  
Souad Amezianne Hassani ◽  
Salah-Eddine Kabbaj ◽  
James S. Okon ◽  
J. Paul Vicknair
Keyword(s):  
Local Ring ◽  
Group Rings
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