University of KashanMathematics Interdisciplinary Research2538-36399120240301Some Remarks on the Annihilating-Ideal Graph of Commutative Ring with Respect to an Ideal11112911421310.22052/mir.2022.246685.1366ENZahra MahmudiankoruieDepartment of mathematics, Faculty of science, University of Qom, I. R. Iran0000-0001-8941-3423Mohammad Hasan NaderiDepartment of mathematics, Faculty of science, University of Qom, I. R. Iran0000000167437619Journal Article20220728The 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 $R$.<br />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}<br />that $ AG_I ( R ) $ is bipartite if and only if $ AG_I ( R ) $ is triangle-free.https://mir.kashanu.ac.ir/article_114213_ca79a921ccca14b6eca0d29cc33196df.pdf