Let R be an associative ring with identity. An element x ∈ R is said to be weakly clean if x = u + e or x = u − e for some unit u and idempotent e in R. The ring R is said to be weakly clean if all of its elements are weakly clean. In this paper we obtain an element-wise characterization of abelian weakly clean rings. A relation between unit regular rings and weakly clean rings is also obtained.

Let R be an associative ring with identity and let J(R) denote the Jacobson radical of R . We say that R is primary if R/J(R) is simple Artinian and J(R) is nilpotent. In this paper we obtain necessary and sufficient conditions for the group ring RG , where G is a nontrivial abelian group, to be primary.

Abstract. Let R be an associative ring with identity and let n be a positive integer.

Let R be an associative ring with identity. An element x ∈ R is clean if x can be written as the sum of a unit and an idempotent in R. R is said to be clean if all of its elements are clean. Let n be a positive integer. An element x ∈ R is n-clean if it can be written as the sum of an idempotent and n units in R. R is said to be n-clean if all of its elements are n-clean. In this paper we obtain conditions which are necessary or sufficient for a group ring to be n-clean.

Let be a ring with identity and let () be a polynomial in ()[ ] where () denotes the center of. An element ∈ is called ()-clean if = + for some , ∈ such that is a unit and () = 0. The ring is ()-clean if every element of is ()-clean. We consider () = (−) where is a unit in such that every root of () is central in. We show, via set-theoretic topology, that among conditions equivalent to being ()-clean, is that is right (left)-topologically boolean.