On Generalized Rickart $\ast$-rings

Abstract

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 $x$. In this work, we introduce generalized right projection of an element in a $\ast$-ring and prove that every element in a generalized Rickart $\ast$-ring has generalized right projection. Various characterizations of generalized Rickart $\ast$-rings are obtained. We introduce the concept of generalized weakly Rickart $\ast$-ring and provide a characterization of generalized Rickart $\ast$-rings in terms of weakly generalized Rickart $\ast$-rings. It is shown that generalized Rickart $\ast$-rings satisfy the parallelogram law. A sufficient condition is established for partial comparability in generalized Rickart $\ast$-rings. Furthermore, it is proved that pair of projections in a generalized Rickart $\ast$-ring possess orthogonal decomposition.

14 pages