$\pi$-BAER $\ast$-RINGS

Abstract

A $\ast$-ring
 $R$ is called a $\pi$-Baer $\ast$-ring, if for any projection invariant left ideal $Y$ of $R$, the right annihilator of $Y $
 is generated, as a right ideal, by a projection.
 In this note, we
 study some properties of such $\ast$-rings.
 We indicate interrelationships between the $\pi$-Baer $\ast$-rings and related classes of rings such as
 $\pi$-Baer rings, Baer $\ast$-rings, and quasi-Baer $\ast$-rings. We announce several
 results on $\pi$-Baer $\ast$-rings.
 We show that this notion is well-behaved with respect to
 polynomial extensions and full matrix rings.
 Examples are provided to explain and delimit our results.