Some Remarks on the Annihilating-Ideal Graph of Commutative Ring with Respect to an Ideal

Abstract

‎The graph $ AG ( R ) $ {of} a commutative ring $R$ with identity has an edge linking two unique vertices when the product of the vertices equals {the} zero ideal and its vertices are the nonzero annihilating ideals of $R$‎.‎The annihilating-ideal graph with {respect to} an ideal $ ( I ) $, ‎which is {denoted} by $ AG_I ( R ) $‎, ‎has distinct vertices $ K $ and $ J $ that are adjacent if and only if $ KJ\subseteq I $‎. ‎Its vertices are $ \{K\mid KJ\subseteq I\ \text{for some ideal}\ J \ \text{and}\ K$‎, ‎$J \nsubseteq I‎, ‎K\ \text{is a ideal of}\ R\} $‎. ‎The study of the two graphs $ AG_I ( R ) $ and $ AG(R/I) $ and {extending certain} prior findings are two main objectives of this research‎. ‎This studys {among other things‎, ‎the} findings {of this study reveal}‎‎that $ AG_I ( R ) $ is bipartite if and only if $ AG_I ( R ) $ is triangle-free‎.

Keywords

• annihilating-ideal graph‎
• ‎cut-point‎
• ‎girth‎
• ‎$ r-$partite graph‎