On generalized zero-divisor graphs of a non-commutative ring with respect to an ideal
Let [Formula: see text] be a non-commutative ring, and [Formula: see text] be an ideal of [Formula: see text]. In this paper, we generalize the definition of the zero-divisor graph of [Formula: see text] with respect to [Formula: see text], and define several generalized zero-divisor graphs of [Formula: see text] with respect to [Formula: see text]. In this paper, we investigate the ring-theoretic properties of [Formula: see text] and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.