关于右RIC环

On Right RIC Rings

本文的主要目的是讨论Sm ith 提出的两个公开问题。这两个问题是:(1) 右PCI环是否为右SI环? (2) 右RIC环右CEPI环和右CEPI环右SI环是否成立? 本文对上述两个公开问题给出了肯定的回答,并且证明了:(1) 设R是右RIC环,若M1 是CS模,M2 是半单模,则M1M2 是CS模;(2) 设R是左Noether右RIC环,则每个有限生成右R-模满足限制极小条件。

In this paper, the author discuss P.F.Smith′s two open problems:(1)Whether right PCI rings are right SI rings.(2) Whether right RIC ringright CEPI ring and right CEPI ringright SI ring are true.The paper answered P.F.Smith′s two open problems affiruatively and showed the following results:(1) Let R be a RIC ring, if M 1 is a CS right R module, M 2 is a semisimple right R module, then M 1M 2 is a CS right R module.(2) Let R be a left noetherian right RIC ring, then every finite generated right R module satisfies restricted minimum condition.