Feebly r-clean ring and feebly*-r-clean ring

Abstract

In this article, we introduce the concept of feebly r-clean ring and feebly ∗-r-clean ring. A ring R is defined to be feebly r-clean, if every element a can be written as a = r + e − f, where u is a regular and e, f are orthogonal idempotents and A ∗ ring R is defined to be feebly ∗-r clean if every element a can be written as a = r + p − q, where r is regular element and p, q are orthogonal projections. Further we generalise this concept of feebly r-clean ring to feebly g(x)−r-clean ring, where g(x) ∈ C(R)[x] and C(R) is the centre of ring R.