BIIDEALS IN INVOLUTION RINGS AND SEMI-GROUPS

Abstract

In a semiprime ring (semigroup with 0), every minimal quasiideal is a minimal biideal and vice versa. Moreover, for prime *-semigroups with 0, each *-minimal *-biideal is a minimal *-biideal. Nevertheless, A *-biideal B of a prime *-ring (*-semigroup) A is minimal if and only if B has the form B = RR∗, for a minimal right ideal R of A. Furthermore, for a semiprime *-semigroup S, each *-minimal *-biideal B is either minimal or a direct union of a minimal biideal C of S and its involutive image C∗. Finally, a set of equivalent conditions are given for a *-simple *-semigroup S to be the union of its *-minimal *-biideals.