IRRELEVANT ATTACHED PRIME IDEALS OF A CERTAIN ARTINIAN MODULE OVER A COMMUTATIVE RING

irrelevant attached prime ideals of a certain artinian module over a commutative ring

let m be an artinian module over the commutative ring a (with nonzero identity) and a p spec a be such that a is a finitely generated ideal of a and am = m. also suppose that h = h where h. = m/ (0: a )for i

Graded Prime Ideals Attached to a Group Graded Module

Let $G$ be a finitely generated abelian group and $M$ be a $G$-graded $A$-module. In general, $G$-associated prime ideals to $M$ may not exist. In this paper, we introduce the concept of $G$-attached prime ideals to $M$ as a generalization of $G$-associated prime ideals which gives a connection between certain $G$-prime ideals and $G$-graded modules over a (not necessarily $G$-graded Noetherian...

Artinian Subrings of a Commutative Ring

Given a commutative ring R, we investigate the structure of the set of Artinian subrings of R . We also consider the family of zero-dimensional subrings of R. Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite interse...

Directed prime graph of non-commutative ring

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

Minimal Prime Ideals of Ore Extensions over Commutative Dedekind Domains

Let R = D[x;σ, δ] be an Ore extension over a commutative Dedekind domain D, where σ is an automorphism on D. In the case δ = 0 Marubayashi et. al. already investigated the class of minimal prime ideals in term of their contraction on the coefficient ring D. In this note we extend this result to a general case δ 6= 0.

Rigid Resolution of a Finitely Generated Module Over a Regular Local Ring