On nilpotent ideals of index 2 in finite commutative rings

Abstract

Let [Formula: see text] be a finite commutative ring with nonzero identity. An ideal [Formula: see text] of [Formula: see text] is called nilpotent of index 2 if [Formula: see text]. [Formula: see text] is called local nilpotent of index 2 if it has a unique maximal nilpotent ideal of index 2. In this paper, we investigate the interplay between cliques in the zero divisor graph and nilpotent ideals of index 2 in [Formula: see text]. In particular, if [Formula: see text] is a maximal clique and [Formula: see text] is an ideal, then [Formula: see text] is a maximal nilpotent ideal of index 2. Also, we study local nilpotent rings of index 2. Finally, we prove that every local ring of order [Formula: see text] with [Formula: see text] is local nilpotent of index 2 and present a counter example for [Formula: see text].