ON YAQUB NIL-CLEAN RINGS

"A ring R is Yaqub nil-clean if a+a3 or a−a3 is nilpotent for all a ∈ R. We prove that a ring R is a Yaqub nil-clean ring if and only if R ∼= R1,R2,R3,R1 ×R2 or R1×R3, where R1/J(R1) is Boolean, R2/J(R2) is a Yaqub ring, R3/J(R3) ∼= Z5 and each J(Ri) is nil, if and only if J(R) is nil and R/J(R) is isomorphic to a Boolean ring R1, a Yaqub ring R2, Z5, R1×R2, or R1×Z5, if and only if for any a ∈ R, there exists e3 = e such that a − e or a + 3e is nilpotent and ae = ea, if and only if R is an exchange Hirano ring. The structure of such rings is thereby completely determined."