Von Neumann正则环和P－V－环

本文中，我们证明了如下结果：（１）环Ｒ是强正则的当且仅当Ｒ是左Ｐ－Ｖ－环且Ｒ的每个极大左理想是拟理想；（２）环Ｒ是强正则的当且仅当Ｒ是半素的且Ｒ的主左理想的极大左次理想是Ｒ的理想，所以有效推广了Ｋａｐｌａｎｓｋｙ的如下结果：可换环Ｒ是ＶｏｎＮｅｕｍａｎｎ正则的当且仅当每一个单Ｒ－模是内射的。

Von Neumann Regular Rings and P-V-Rings

In this paper. we prove the following results: (1) A ring R is strongly regular if and only if R is a left p-V-ring and verey maximal left ideal of R is a quasiideal; (2) A ring R is trongly regular if and only if R is fully semiprime and every maximal left subideal in a principal ideal of R is an ideal. Therefore. the following important theorem of Kaplansky is effectively extended: A commutative ring R is von Neumann regular if and only if every simple R -module is injective. Further properties of p-V-rings are also considered .