There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Commutative rings I

Let L(R) denote the set of all non-trivial left ideals of a ring R. The intersection graph of ideals of a ring R is an undirected simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with L(R) and two distinct vertices are joined by an edge if and only if the corresponding left ideals of R have a non-zero intersection. The ideal structure of a ring reflects many ring theoretical properties. Thus much research has been conducted last few years to explore the properties of G(R). This is a survey of the developments in the study on the intersection graphs of ideals of rings since its introduction in 2009.

http://comb-opt.azaruniv.ac.ir/article_14242_28ea32a9e9606d73a6867e018edd58df.pdf

A survey on the Intersection graphs of ideals of rings

Let Rbe a commutative ring with non-zero identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r∈R\{0} such that Ir=(0). Let A(R) denote the set of all annihilating ideals of R and A (R)*=A(R)\{0}. In this article, we introduce a new graph associated with R denoted by H(R) whose vertex set is A(R)* and two distinct vertices I, J are adjacent in this graph if and only if IJ=(0) or I+J ∈ A(R). The aim of this article is to study the interplay between the ring-theoretic properties of a ring R and the graph-theoretic properties of H(R). For such a ring R, we prove that H(R) is connected and find its diameter. Moreover, we determine girth of H(R). Furthermore, we provide some sufficient conditions under which H(R) is a complete graph.

/pdf/on-the-extended-graph-associated-with-the-set-of-all-non-2le563sa.pdf

On the extended graph associated with the set of all non-zero annihilating ideals of a commutative ring

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal.   Let $R$ be a ring. Let us denote  the collection  of all proper ideals  of $R$ by $mathbb{I}(R)$  and $mathbb{I}(R)backslash {(0)}$ by $mathbb{I}(R)^{*}$.  With $R$, we associate an undirected graph denoted by $g(R)$, whose vertex set is $mathbb{I}(R)^{*}$ and distinct vertices $I_{1}, I_{2}$ are adjacent in $g(R)$  if and only if $I_{1}cap I_{2}neq I_{1}I_{2}$.  The aim of this article is to study the interplay between the graph-theoretic properties of $g(R)$ and the ring-theoretic properties of $R$.

https://jart.guilan.ac.ir/article_3328_37da989245b3ff3ca164523e990de30b.pdf

Some results on a subgraph of the intersection graph of ideals of a commutative ring

The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$,  we mean an ideal $I$ of $R$ such that $Ineq R$.  We say that a proper ideal $I$ of a ring $R$ is a  maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$, $I$ is maximal with respect to the property of being not a prime ideal.  The concept of maximal non-maximal ideal and maximal non-primary ideal of a ring can be similarly defined. The aim of this article is to characterize ideals $I$ of a ring $R$ such that $I$ is a maximal non-prime (respectively, a maximal non maximal, a maximal non-primary) ideal of $R$.

https://jart.guilan.ac.ir/article_1213_ef7c4b9f1125da4eb7d276b1d9cbcb6d.pdf

A note on maximal non-prime ideals

Let R be a commutative ring with identity which is not an integral domain. Let A(R)∗ denote the collection of all nonzero annihilating ideals of R and AG(R) denote the annihilating ideal graph of R. In this article, we consider the dominating sets of (AG(R))c (where (AG(R))c is the complement of AG(R)) and study the influence of the dominating sets of (AG(R))c on the ring structure of R and vice-versa.

On the dominating sets of the complement of the annihilating ideal graph of a commutative ring

The rings considered in this article are commutative with identity. This article is motivated by the work on comaximal graphs of rings.  In this article, with any ring $R$, we associate an undirected graph denoted by $G(R)$, whose vertex set is the set of all elements of $R$ and distinct vertices $x,y$ are joined by an edge in $G(R)$ if and only if $Rxcap Ry = Rxy$.  In Section 2 of this article, we classify rings $R$ such that $G(R)$ is complete and we also consider the problem of determining rings $R$ such that $chi(G(R)) = omega(G(R))< infty$. In Section 3 of this article, we classify rings $R$ such that $G(R)$ is planar.

https://jart.guilan.ac.ir/article_2399_1591422012d4680950f3bb1760b533a6.pdf

S. Visweswaran

Papers

Some results on a subgraph of the intersection graph of ideals of a commutative ring

A note on maximal non-prime ideals

On the dominating sets of the complement of the annihilating ideal graph of a commutative ring

A Note on a graph associated to a commutative ring