For semiprime involution rings, we determine some ∗-minimal ∗-ideals using idempotent elements. Nevertheless, ∗-minimal ∗-biideals are characterized by idempotent elements. Moreover, the involutive version of a theorem due to Steinfeld, which investigates a semiprime involution ring A if A=SocA, is given. Finally, semiprime involution rings having no proper nonzero ∗-biideals are characterized.
The consistent way of investigating rings with involution, briefly *-rings, is to study them in the category of *-rings with morphisms preserving also involution. In this paper we continue the study of *-rings and the notion of *-reduced *-rings is introduced and their properties are studied. We introduce also the class of *-Baer *-rings. This class is defined in terms of *-annihilators and principal *-biideals, and it naturally extends the class of Baer *-rings. The use of *-biideals makes this concept more consistent with the involution than the use of right ideals in the notion of Baer *-rings. We prove that each *-Baer *-ring is semiprime. Moreover, we show that the property of *-Baer extends to both the *-corner and the center of the *-ring. Finally, we discuss the relation between *-Baer and quasi-Baer *-rings; the generalization of Baer *-ring.