Zero-divisor graphs of idealizations

Abstract

We consider zero-divisor graphs of idealizations of commutative rings. Specifically, we look at the preservation, or lack thereof, of the diameter and girth of the zero-divisor graph of a ring when extending to idealizations of the ring.

Total Graphs of Idealization

“…In [5] D.F.Anderson, A.Badawi studied connectedness of Total graph of the idealization R(+)M and also investigate diameter and has proved some results on girth of Total graphs. Different aspects of the idealization are thoroughly investigated in [10], [11]. In this paper also extend the study of D.F.Anderson, and A.Badawi .…”

Extended Zero-Divisor Graphs of Idealizations

“…As in the classical case [11], if |M | ≥ 4, then gr(Γ(R ⋉ M )) = 3, since (0, m 1 ) − (0, m 2 ) − (0, m 3 ) − (0, m 1 ) is a cycle of length three, where m 1 , m 2 and m 3 are distinct nonzero elements of M . Thus, we only need to consider when the module M has only two or three elements.…”

“…It was Anderson and Livingston (in [6]) who introduced the zero-divisor graph of a commutative ring and started the study of the relationship between ring-theoretic properties and graph-theoretic ones. Since then, zero-divisor graphs of commutative rings have attracted the attention of several researchers (see, for instance, [1,3,4,5,6,7,10,11,16,17]). It was proved, among other things, that Γ(R) is connected with diam(Γ(R)) ≤ 3 and gr(Γ(R)) ∈ {3, 4, ∞}.…”

“…In order to well understand the concept of zero-divisor graphs, several authors have been interested in studying zero-divisor graphs of particular ring constructions (see for instance [10,11]). In this paper, we are interested in studying extended zero-divisor graphs of idealizations.…”

Zero-divisor graphs of idealizations with respect to prime modules

“…The zero-divisor graph of a commutative ring has been studied extensively by several authors, e.g. [1,2,4]. Our aim in this note is to study the diameter and girth of the zero-divisor graph of the ring R(+)M, where M is a prime R-module.…”