关于正则环的几点注记

SOME REMARKS ON REGULAR RING

环Ｒ称为左（右）ＳＦ）环，如果所有单左（右）Ｒ－模是平坦的；环Ｒ称为Ｉ－环，如果Ｒ的每个非零左理想含有非零幂等元，在本文中，我们证明了如下主要结果：（－）对于环Ｒ，如下条件是等价的：（１）Ｒ是Ａｒｔｉｎ半单环；（２）Ｒ是左ＳＦ－环且Ｒ／Ｚ（ＲＲ）是Ａｒｔｉｎ半单环；（３）Ｒ是左非奇异的，左ＳＦ－环且ＲＲ具有有限秩；（４）Ｒ是正交有限的Ｉ－环。（二）Ｒ是基层不为零的正则左自内射环当且仅当Ｒ是包含非奇异内射极大左理想的左Ｐ－内射环。

A ring R is called a left(right)SF-ring if every simple left(right)R-module is flat.A ring R is called an I-ring if every nonzero felt ideal of R contains a nonzero idempotent.In this paper, we prove the following results:(1)For a ring R, the following conditions are equivalent:(i)R is an Artinian semisimple ring;(ii) R is a left SF-ring and R/Z(RR) is an Artinian semisimple ring;(iii) R is a left nonsingular, left SF-ring and RR has finite rank;(iv) R is an orthogonally finite I-ring.(2) R is a left selfinjective regular ring with nonzero socle if and only if R is a left p-injective ring containing a nonsingular injective maximal left ideal.