文摘
In this paper, we associate an undirected graph e503b" title="Click to view the MathML source">AG(S), the annihilating-ideal graph, to a commutative semigroup S . This graph has vertex set bb1dd27" title="Click to view the MathML source">A⁎(S)=A(S)∖{(0)}, where A(S) is the set of proper ideals of S with nonzero annihilator. Two distinct vertices b911e4741b398190" title="Click to view the MathML source">I,J∈A⁎(S) are defined to be adjacent in e503b" title="Click to view the MathML source">AG(S) if and only if e8d7f321671f97f0f" title="Click to view the MathML source">IJ=(0), the zero ideal. Conditions are given to ensure a finite graph. Semigroups for which each nonzero, proper ideal of S is an element of bbfd039685788f340bb546d3ce89" title="Click to view the MathML source">A⁎(S) are characterized. Connections are drawn between e503b" title="Click to view the MathML source">AG(S) and e8fc213e87201afc6f20cf84826d8" title="Click to view the MathML source">Γ(S), the well-known zero-divisor graph, and the connectivity, diameter, and girth of e503b" title="Click to view the MathML source">AG(S) are described. Semigroups S for which e503b" title="Click to view the MathML source">AG(S) is a complete or star graph are characterized. Finally, it is proven that the chromatic number is equal to the clique number of the annihilating ideal graph for each reduced semigroup and null semigroup. Upper and lower bounds for bee6fc" title="Click to view the MathML source">χ(AG(S)) are given for a general commutative semigroup.