Square-zero determined matrix algebras

Abstract

Let N n (R) be the algebra of all n × n strictly upper triangular matrices over a commutative unital ring R. It is shown in this article that N n (R) is square-zero determined. More definitely, if a symmetric bilinear map φ from N n (R) × N n (R) to an R-module V satisfies the condition that φ(u, u) = 0 whenever u 2 = 0, then there exists a linear map ϕ from to V such that φ(x, y) = ϕ(xy + yx) for all x, y in N n (R). As applications of this result, we show that (i) a linear map on N n (R) is a square-zero derivation if and only if it is a quasi Jordan derivation; (ii) an invertible linear map on N n (R) is a square-zero preserving map if and only if it is a quasi Jordan automorphism.