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

Let R be a commutative ring with A( R) its set of annihilating-ideals. The extended annihilating-ideal graph of R, denoted by (AG) over bar (R), is an undirected graph with vertex set A(R)* = A(R) \ {0} and two vertices I-1 and I-2 are adjacent if and only if I-1(m) I-2(n) = 0 with I-1(m) not equal 0 and I-2(n) not equal 0, for some positive integers m and n. In this paper, we first study some basic properties of (AG) over bar (R) and then we investigate the relationship between the extended annihilating-ideal graph (AG) over bar (R), the annihilator-ideal graph A(I)(R) and the annihilating-ideal graph AG(R) of a commutative ring R.