SZ-Derivations, PZ-Derivations and S-Derivations of a Matrix Algebra over Commutative Rings

Abstract:

      Let $R$ be a commutative ring with identity, $N_n(R)$ the matrix algebra consisting of all $n\times n$ strictly upper triangular matrices over $R$ with the usual product operation. An $R$-linear map $\phi: N_n(R)\to N_n(R) $ is said to be an SZ-derivation of $N_n(R)$ if $x^2=0$ implies that $\phi(x)x x\phi(x)=0$. It is said to be an S-derivation of $N_n(R)$ if $\phi(x^2)=\phi(x)x x\phi(x)$ for any $x\in N_n(R)$. It is said to be a PZ-derivation of $N_n(R)$ if $xy=0$ implies that $\phi(x)y x\phi(y)=0$. In this paper, by constructing several types of standard SZ-derivations of $N_n(R)$, we first characterize all SZ-derivations of $N_n(R)$. Then, as its application, we determine all S-derivations and PZ-derivations of $N_n(R)$, respectively.