In this paper, we first show that the quotient ring Z[x]/(x 2 + x) is an involution-ring with the involution * given by (a 1 + a 2 x) * = a 1 − a 2 − a 2 x, where a 1 , a 2 ∈ Z. Then, we determine explicitly all invertible elements, regular elements, MP-inverses, group invertible elements, EP elements and SEP elements of Z[x]/(x 2 + x). Furthermore, we give a new characterization for Abel rings.