On two extensions of the annihilating-ideal graph of commutative rings

Abstract

Let R be a commutative ring with A ⁢ ( R ) {A(R)} its set of annihilating-ideals. The extended annihilating-ideal graph of R, denoted by AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} , is an undirected graph with vertex set A ⁢ ( R ) * = A ⁢ ( R ) ∖ { 0 } {A(R)^{*}=A(R)\setminus\{0\}} and two vertices I 1 {I_{1}} and I 2 {I_{2}} are adjacent if and only if I 1 m ⁢ I 2 n = 0 {I_{1}^{m}I_{2}^{n}=0} with I 1 m ≠ 0 {I_{1}^{m}\neq 0} and I 2 n ≠ 0 {I_{2}^{n}\neq 0} , for some positive integers m and n. In this paper, we first study some basic properties of AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} and then we investigate the relationship between the extended annihilating-ideal graph AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} , the annihilator-ideal graph A I ⁢ ( R ) {A_{I}(R)} and the annihilating-ideal graph AG ⁢ ( R ) {\mathrm{AG}(R)} of a commutative ring R.