On Weakly K-Clean Rings

Abstract

In this paper, we offer a new generalization of the k-clean ring that is called weakly k-clean ring. Let 2 ≤ k ∈ N. Then the ring R is said to be a weakly k-clean if for each a ∈ R there exist u ∈ U(R) and e ∈ Pk(R) such that a = u+e or a =u−e. We obtain some properties of weakly k-clean rings. It is shown that each homomorphic image of a weakly k-clean ring is weakly k-clean. Also, it is proved that the ring R[R,S] is weakly k-clean if and only if R is k-clean and S is weakly k-clean. 

Key words and phrases. weakly k-clean ring, k-clean ring.

2010 Mathematics Subject Classification. 16U99, 16Z05