The existence of clean element and feebly clean element in a matrix ring

Abstract

A ring R with unity is called clean if every element x ∈ R can be written as a sum of an idempotent and a unit element in ring R. Meanwhile, a ring with unity is called feebly clean, if every element r in the ring can be written as r = u + e − f, with ef = fe = 0, where u is a unit element in the ring and e, f is an idempotent element in the ring. [7] shows the existence of clean elements in a subring X3(D) of the matrix ring 3 × 3 over an integral domain D. Based on these results, we provide the set of all clean elements in the matrix ring X3(ℤ), we show the existence of feebly clean elements in the matrix ring X3(ℤ), and show the connections between those elements and feebly clean elements in the matrix ring X3(ℤ). These connections are different from [10].