Generalizations of Semiregular Rings^#

^#Communicated by Alberto Facchini.

In this article, we call a ring R right generalized semiregular if for any a ∈ R there exist two left ideals P, L of R such that lr(a) = P⊕ L, where P ⊆ Ra and Ra ∩ L is small in R. The class of generalized semiregular rings contains all semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-π-regular if for any a ∈ R there exist a positive integer n and e ² = e ∈ a ⁿ R such that (1 − e)a ⁿ  ∈ J(R), the Jacobson radical of R. It is shown that a ring R is semi-π-regular if and only if R/J(R) is π-regular and idempotents can be lifted modulo J(R).

Keywords:

• Exchange rings
• AP-injective rings
• Semi-π-regular rings
• Generalized semiregular rings

2000 Mathematics Subject Classification:

• 16D50
• 16E40
• 16D40
• 16A30

ACKNOWLEDGMENTS

The first author would like to thank the referee for his/her valuable suggestions. He also would like to thank Professor Nanqing Ding and Doctor Weixing Chen for their useful advice. This work is supported by the Doctorate Foundation of China Education Ministry (Grant No. 20020284009).

Notes

^#Communicated by Alberto Facchini.