关于Goldie环的条件的一点注记

A NOTE ON THE DEFINITION OF GOLDIE RINGS

<正> 在[1][2]中许永华对结合环R引入右R-模同态链归纳条件,可以叙述为:设r∈R,令元素r的右零化子r~⊥={x∈R|rx=0}。设M={r~⊥,AOr∈R},则{M,}作成一个偏序集。我们说结合环R满足右R-模同态链归纳条件,如果偏序集{M,}中每一链 (即M的有序子集) 都有最小上界。[3]中对环R引入了一个较之弱的条件,我们将称之为特殊右零化子集归纳条件,这是指,要求偏序集{M,}是个归纳集。

It is said that a ring R satisfies inductive condition for the set of special right annihilators, if the partially ordered set {M,}, where M={r~⊥,0≠r∈R} and r⊥= {x∈R|rx=0}, is inductive. This paper proves the following Theorem. Let R be a semi-prime ring. If R has finite right Goldie dimension and satisfies the inductive condition for the set of special right annihilators, then R is right Goldie ring.