交换环上的严格上三角代数上的双导子

Biderivations of the Algebra of Strictly Upper Triangular Matrices over a Commutative Ring

中文关键词:  双导子  严格上三角矩阵  代数.

英文关键词:biderivation  strictly upper triangular matrix  algebra.

中文摘要:

      设$N_n(R)$是带有单位元的交换环$R$上的$n\times n$严格上三角矩阵构成的代数.$R$-双线性映射$\phi :N_n(R)\times N_n(R)\longrightarrow N_n(R)$称为双导子是指它关于每一个变元是导子.本文给出了$N_n(R)$上的中心双导子和极端双导子的定义,证明了当$n\geq 5$时,$N_n(R)$每一个双导子可分解为一个内双导子,一个中心双导子和和一个极端双导子之和.

英文摘要:

      Let $N_n(R)$ be the algebra consisting of all strictly upper triangular $n\times n$ matrices over a commutative ring $R$ with the identity. An $R$-bilinear map $\phi :N_n(R)\times N_n(R)\longrightarrow N_n(R)$ is called a biderivation if it is a derivation with respect to both arguments. In this paper, we define the notions of central biderivation and extremal biderivation of $N_n(R)$, and prove that any biderivation of $N_n(R)$ can be decomposed as a sum of an inner biderivation, central biderivation and extremal biderivation for $n\geq 5$.