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.
Acknowledgements
We are grateful to the referee for providing us with substantial help.