Ali Shahidikia - Department of Mathematics, Dezful Branch, Islamic Azad University, Dezful, Iran.
Haimd Haj Seyyed Javadi - Department of Computer Engineering, Shahed University, Tehran, Iran.

‎A *-ring‎ ‎R is called a generalized \pi-Baer *-ring‎, ‎if for any projection invariant left ideal Y of R‎, ‎the right annihilator of Y^n‎ ‎is generated‎, ‎as a right ideal‎, ‎by a projection, ‎for some positive integer n‎, ‎depending on Y‎. ‎In this paper, ‎we‎ ‎study some properties of generalized \pi-Baer *-rings‎. ‎We show that this notion is well-behaved with respect to‎ ‎polynomial extensions, full matrix rings, and several classes of triangular matrix rings‎.‎ ‎We indicate interrelationships between the generalized \pi-Baer *-rings and related classes of rings such as‎ generalized ‎\pi-Baer rings‎, generalized ‎Baer *-rings‎, generalized quasi-Baer *-rings, and ‎‎\pi-Baer \s-rings. ‎We obtain algebraic examples which are generalized‎ ‎\pi-Baer \ast -rings but are not ‎\pi-Baer *-rings‎. ‎We show that for pre-C*-algebras these two notions are equivalent‎.‎We obtain classes of Banach *-algebras‎ ‎which are generalized‎ ‎\pi-Baer *-rings but are not ‎\pi-Baer *-rings‎. We finish the paper by showing that for a locally compact‎‎abelian group G‎, ‎the group algebra L^{۱}(G) is a‎ ‎generalized ‎\pi-Baer *-ring‎, ‎if and only if so is the group C*-algebra‎ ‎C^{*}(G)‎, ‎if and only if G is finite‎.

generalized π-Baer ring, generalized π-Baer ∗-ring, generalized quasi-Baer ∗-ring, generalized Baer ring, generalized Baer ∗-ring