可换环上矩阵代数的SZ-导子, PZ-导子 和 S-导子

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

中文关键词:  导子  SZ-导子  PZ-导子  S-导子.

英文关键词:SZ-derivations  S-derivations  PZ-derivations.

中文摘要:

      设$R$是可换环, $N_n(R)是所有严格上三角矩阵组成的矩阵代数. 本文决定了$N_n(R)$上的所有SZ-导子, PZ-导子和S-导子.

英文摘要:

      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.