Weakly unit regular clean rings

A ring R is called clean if every member of R is the sum of a self-resolved member and an invertible member. Also, we call the ring R weakly clean if every member of R can be written as the sum or difference of an invertible member and an autoregressive member. The a\in R member is called unit regular whenever u\in U(R) exists such a=aua. A ring R is called a clean unit if every member of R is the sum of an autonomial member and a unitary member. We call a ring R a weakly clean unit if every member of R can be written as the sum or difference of a unit regular and a self-power term. In this paper, weakly unit regular clean rings are introduced and discussed. In particular, we show that if \{R_i\}_{i\in I} is a family of commutative rings, then R=\prod_{i\in I} R_i is weakly unit regular clean if and only if every R_i is weakly clean regular unit and at most one R_i's are not clean and regular units.

کلیدواژه ها:

clean rings ، weakly clean rings ، unit clean rings ، weakly clean regular rings