Regular Elements and BQ-Elements of the Semigroup (Zn, ·)

A semigroup S is called a BQ-semigroup if the sets of bi-ideals and quasi-ideals of S coincide. By a BQ-element of S we mean an element x ∈ S such that the bi-ideal and the quasi-ideal of S generated by x coincide. It is known that every regular semigroup is a BQ-semigroup. We also have that every regular element of S is a BQ-element. We have known that (Zn, ·), the multiplicative semigroup of integers modulo n, is a regular semigroup if and only if n is a square-free, and it is a BQ-semigroup if and only if either n = 4 or n is square-free. In this paper, the regular elements and the BQ-elements of (Zn, ·) are characterized and we show that the above known results become consequences of these characterizations