Ideal neighbourhoods in a ring

The annihilator-inclusion Ideal graph of a commutative ring

Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected grap...

The sum-annihilating essential ideal graph of a commutative ring

Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$is called an annihilating ideal if there exists $rin Rsetminus {0}$ such that $Ir=(0)$ and an ideal $I$ of$R$ is called an essential ideal if $I$ has non-zero intersectionwith every other non-zero ideal of $R$. Thesum-annihilating essential ideal graph of $R$, denoted by $mathcal{AE}_R$, isa graph whose vertex set is the set...

Notes on amalgamated duplication of a ring along an ideal

In this paper, we study some ring theoretic properties of the amalgamated duplication ring $Rbowtie I$ of a commutative Noetherian ring $R$ along an ideal $I$ of $R$ which was introduced by D'Anna and Fontana. Indeed, it is determined that when $Rbowtie I$ satisfies Serre's conditions $(R_n)$ and $(S_n)$, and when is a normal ring, a generalized Cohen-Macaulay ring and finally a filter ring.

A note on a graph related to the comaximal ideal graph of a commutative ring

  ‎The rings considered in this article are commutative with identity which admit at least two maximal ideals‎.  ‎This article is inspired by the work done on the comaximal ideal graph of a commutative ring‎. ‎Let R be a ring‎.  ‎We associate an undirected graph to R denoted by mathcal{G}(R)‎,  ‎whose vertex set is the set of all proper ideals I of R such that Inotsubseteq J(R)‎, ‎where J(R) is...

The Maximal Regular Ideal of a Ring