On two extensions of the classical zero-divisor graph
Abstract: The extended zero-divisor graph and the annihilator graph of a ring are two extensions of the classical zero-divisor graph. In this paper, we investigate the relation between these graphs. Relation between these graphs on some particular ring constructions is also given.
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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.
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.
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