On maps preserving square-zero matrices

Abstract

The problem of characterizing linear operators on matrix algebras that leave invariant certain functions, subsets or relations has attracted the attention of many mathematicians (see survey papers [20–22,27] for details). For instance, linear operators preserving zeroproduct of matrices are studied in [15,16,29,31]; linear operators preserving idempotent matrices are studied in [1,11,13,14]; linear operators preserving matrices annihilated by a fixed polynomial are studied in [12,13,18]; linear operators preserving nilpotent matrices are studied in [9]; and linear operators preserving square-zero matrices are studied in [29]. Most linear preserver problems were investigated in the case of matrix algebras over fields. In contrast, not too much is known about matrix algebras over commutative rings. To the best of our knowledge, beside the papers by McDonald [25] and Waterhouse [30], only Bresar and Semrl [11] used elementary calculations to describe linear maps preserving idempotent matrices over commutative rings. The reason why people seldom study matrix algebras over commutative rings is probably that one might encounter some difficulties that are not easy to overcome. For example, unlike matrix algebras over fields, the automorphisms on matrix algebras over commutative rings can fail to be inner (see [19,28]).