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A graph associated with the set of all nonzero annihilating ideals of a commutative ring

S. Visweswaran

Department of Mathematics, Saurashtra University, Rajkot, 360005, Gujarat, India

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and

Hiren D. Patel

Department of Mathematics, Saurashtra University, Rajkot, 360005, Gujarat, India

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https://doi.org/10.1142/S1793830914500475Cited by:20 (Source: Crossref)

Abstract

Let R be a commutative ring with 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). In this paper, we consider a simple undirected graph associated with R denoted by Ω(R) whose vertex set equals the set of all nonzero annihilating ideals of R and two distinct vertices I, J in this graph are joined by an edge if and only if I + J is also an annihilating ideal of R. In this paper, for any ring R which is not an integral domain, the problem of when Ω(R) is connected is discussed and if Ω(R) is connected, then it is shown that diam(Ω(R)) ≤ 2. Moreover, it is verified that gr(Ω(R)) ∈ {3, ∞}. Furthermore, rings R such that ω(Ω(R)) < ∞ are characterized.

Keywords:

AMSC: 13A15

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