环上矩阵环的导子

文1]讨论了除环上2阶全矩阵环的导子的一些性质,本文继此讨论一般结合环R上的R阶全矩阵环R_n的导子的性质.环R的加群自同态(?)称为R的导子,若对x、y∈R,有d(xy)=xd(y) d(x)y.如下总假定R有单位元,且用R_n表示R上的n阶全矩阵环,E_ij表示(i,j)位置元素为R的单位元1其余元素为零的R_n的矩阵单位,xE饰表示对角线上元素为x的数量阵.

Derivations of the Full Matrix Rings over a Ring

Let R be a ring with identity. Let R_n denote the full matrix ring over R and d be a derivation of R_n. In this note, it is shown that each derivation d of R_n may determine a derivation f of R such that d(xE) can be expressed by f for every x in R(E be the unity of R_n). Moreover, the relations between d-and f are discussed. In particulat, a necessary condition for a derivation of R to be extendable to one of R_n's is given.