Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct...Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.展开更多
文摘Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.
