A ring R is almost unit-clean provided that every element in R is equivalent to the sum of an idempotent and a regular element. We prove that every ring in which every zero-divisor is strongly pi-regular is almost unit-clean ...

An ideal I of a ring R is strongly pi-regular if for any x is an element of I there exist n is an element of N and y is an element of I such that x(n) x(n+l)y. We prove that every strongly pi-regular ideal of a ring is a ...

An exchange ideal I of a ring R is locally comparable if for every regular x is an element of I there exists a right or left invertible u is an element of 1 + I such that x = xux. We prove that every matrix extension of ...

A *-ring R is called a medium *-clean ring if every element in R is the sum or difference of an element in its Jacobson radical and a projection that commute. We prove that a ring R is medium *-clean if and only if R is ...

A ring R is uniquely (strongly) clean provided that for any a ∈ R there exists a unique idempotent e ∈ R ( e ∈ comm(a) ) such that a − e ∈ U(R). We prove, in this note, that a ring R is uniquely ...