SZ-Derivations, PZ-Derivations and S-Derivations of a Matrix Algebra over Commutative Rings
Received:March 18, 2008  Revised:July 07, 2008
Key Words: SZ-derivations   S-derivations   PZ-derivations.
Fund Project:Fond of China University of Mining and Technology.
 Author Name Affiliation WANG Xian Department of Mathematics, China University of Mining and Technology, Jiangsu 221008, China Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan WANG Deng Yin Department of Mathematics, China University of Mining and Technology, Jiangsu 221008, China
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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.