Reduced zerodivisor graphs of posets

This paper investigates properties of the reduced zerodivisor graph of a poset. We show that a vertex is an annihilator prime ideal if and only if it is adjacent to all other annihilator prime ideals and there are always two annihilator prime ideals which are not adjacent to a nonannihilator prime ideal. We also classify all posets whose reduced zerodivisor graph is planar or toroidal and the number of distinct annihilator prime ideals is four or seven.