Characterizations of the Commutators Involving Idempotents in Certain Subrings of $m_{2}(\\mathbb{z})$

Abstract

In this paper, we characterize the idempotency, cleanness, and unit-regularity of the commutator $[E_1, E_2]$ = $E_1E_2 − E_2E_1$ involving idempotents $E_1, E_2$ in certain subrings of $M_2(Z)$.