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

	Some results on a subgraph of the intersection graph of ideals of a commutative ring
Journal of Algebra and Related Topics
مقاله 3، دوره 6، شماره 2، اسفند 2018، صفحه 35-61 اصل مقاله (391.98 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22124/jart.2018.11188.1114
نویسندگان
S. Visweswaran^* ¹؛ P. Vadhel²
¹Department of Mathematics, Saurashtra University, Rajkot, India.
²Department of Mathematics, Saurashtra University, Rajkot, India
چکیده
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 $R$.
کلیدواژه‌ها
Artinian ring؛ Special principal ideal ring؛ diameter؛ girth؛ clique number

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