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Original Articles

Square-zero determined matrix algebras

, &
Pages 1311-1317 | Received 15 Oct 2010, Accepted 08 Nov 2010, Published online: 14 Apr 2011
 

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.

AMS Subject Classifications:

Acknowledgements

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

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