Author/Authors :
Shahidikia ، Ali Department of Mathematics - Islamic Azad University, Dezful Branch , Haj Seyyed Javadi ، Haimd Department of Computer Engineering - Shahed University
Abstract :
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 $L^{1}(G)$ 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.
Keywords :
generalized Baer ring , generalized π-Baer ring , generalized π-Baer ∗-ring , generalized quasi-Baer ∗-ring , generalized Baer ∗-ring
