查看文章

A ring $ R $ with identity is called``clean''if $~ $ for every element $ ain R $, there exist an idempotent $ e $ and a unit $ u $ in $ R $ such that $ a= u+ e $. Let $ C (R) $ denote the center of a ring $ R $ and $ g (x) $ be a polynomial in $ C (R)[x] $. An element $ rin R $ is called``g (x)-clean''if $ r= u+ s $ where $ g (s)= 0$ and $ u $ is a unit of $ R $ and, $ R $ is $ g (x) $-clean if every element is $ g (x) $-clean. In this paper we define a ring to be weakly $ g (x) $-clean if each element of $ R $ can be written as either the sum or difference of a unit and a root of $ g (x) $.