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The annihilator ideal graph of a commutative ring


Abstract


Let R be a commutative ring with nonzero identity and I be a proper ideal of R. The annihilator graph of R with respect to I, which is denoted by AG_{I}(R), is the undirected graph with vertex-set V(AG_{I}(R)) = \lbrace x\in R \setminus I :   xy \in I\ for some \  y \notin I \rbrace and two distinct vertices x and y are adjacent if and only if A_{I}(xy)\neq A_{I}(x) \cup A_{I}(y), where A_{I}(x) = \lbrace r\in R : rx\in I\rbrace. In this paper, we study some basic properties of AG_I(R), and we characterise when  AG_{I}(R) is planar, outerplanar or a ring graph. Also, we study the graph AG_{I}(\mathbb{Z}_{n}) , where Z_n is the ring of integers modulo n.

DOI Code: 10.1285/i15900932v36n1p1

Keywords: Zero-divisor graph; Annihilator graph; Girth; Planar graph; Outerplanar; Ring graph

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