The Existence of Clean Element and Feebly Clean Element in a Matrix Ring

摘要

A ring R with unity is called clean if every element x E 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 X_3(D) of the matrix ring 3x3 over an integral domain D. Based on these results, we provide the set of all clean elements in the matrix ring X_3(Z), we show the existence of feebly clean elements in the matrix ring X_3(Z), and show the connections between those elements and feebly clean elements in the matrix ring X_3(Z). These connections are different from [10].