Generalized Rickart $\ast$-rings

As a common generalization of Rickart $\ast$-rings and generalized Baer $\ast$-rings, we say that a ring $R$ with an involution $\ast$ is a generalized Rickart $\ast$-ring if for all $x\in R$ the right annihilator of $ x^n$ is generated by a projection for some positive integer $n$ depending on $x$. The abelian generalized Rickart $\ast$-rings are closed under finite direct product. We address the behavior of the generalized Rickart $\ast$ condition with respect to various constructions and extensions, present some families of generalized Rickart $\ast$-rings, study connections to the related classes of rings, and indicate various examples of generalized Rickart $\ast$-rings. Also, we provide some large classes of finite and infinite-dimensional Banach $\ast$-algebras that are generalized Rickart $\ast$-rings but neither Rickart $\ast$-rings nor generalized Baer $\ast$-rings.