The annihilator ideal graph of a commutative ring
Abstract
Let
be a commutative ring with nonzero identity and
be a proper ideal of
. The annihilator graph of
with respect to
, which is denoted by
, is the undirected graph with vertex-set
for some
and two distinct vertices
and
are adjacent if and only if
, where
. In this paper, we study some basic properties of
, and we characterise when
is planar, outerplanar or a ring graph. Also, we study the graph
, where
is the ring of integers modulo
.
![R](http://212.189.136.205/plugins/generic/latexRender/cache/e1e1d3d40573127e9ee0480caf1283d6.png)
![I](http://212.189.136.205/plugins/generic/latexRender/cache/dd7536794b63bf90eccfd37f9b147d7f.png)
![R](http://212.189.136.205/plugins/generic/latexRender/cache/e1e1d3d40573127e9ee0480caf1283d6.png)
![R](http://212.189.136.205/plugins/generic/latexRender/cache/e1e1d3d40573127e9ee0480caf1283d6.png)
![I](http://212.189.136.205/plugins/generic/latexRender/cache/dd7536794b63bf90eccfd37f9b147d7f.png)
![AG_{I}(R)](http://212.189.136.205/plugins/generic/latexRender/cache/d2e7025e0af99deab238086e7ddc201d.png)
![V(AG_{I}(R)) = \lbrace x\in R \setminus I : xy \in I\](http://212.189.136.205/plugins/generic/latexRender/cache/c538ac05933b974a15a8189bf2e073ce.png)
![\ y \notin I \rbrace](http://212.189.136.205/plugins/generic/latexRender/cache/035686148c82b06649e35026763a68bd.png)
![x](http://212.189.136.205/plugins/generic/latexRender/cache/9dd4e461268c8034f5c8564e155c67a6.png)
![y](http://212.189.136.205/plugins/generic/latexRender/cache/415290769594460e2e485922904f345d.png)
![A_{I}(xy)\neq A_{I}(x) \cup A_{I}(y)](http://212.189.136.205/plugins/generic/latexRender/cache/c4c7b5c3d589b9bbd03359a36c300b88.png)
![A_{I}(x) = \lbrace r\in R : rx\in I\rbrace](http://212.189.136.205/plugins/generic/latexRender/cache/695229f7fc8d1f0c500b72e0b694d53c.png)
![AG_I(R)](http://212.189.136.205/plugins/generic/latexRender/cache/b6cc726afbfecd7ac636bd3b1d9a4652.png)
![AG_{I}(R)](http://212.189.136.205/plugins/generic/latexRender/cache/6a77788912fe03cef17b290d5691fe5f.png)
![AG_{I}(\mathbb{Z}_{n})](http://212.189.136.205/plugins/generic/latexRender/cache/6fe2e9809daad6da06ba723328479984.png)
![Z_n](http://212.189.136.205/plugins/generic/latexRender/cache/f7f8f68027bafb6481b1f5c7fa2de658.png)
![n](http://212.189.136.205/plugins/generic/latexRender/cache/7b8b965ad4bca0e41ab51de7b31363a1.png)
DOI Code:
10.1285/i15900932v36n1p1
Keywords:
Zero-divisor graph; Annihilator graph; Girth; Planar graph; Outerplanar; Ring graph
Full Text: PDF