# On Generalized Rickart $*$-rings

## TL;DR

This paper introduces and characterizes generalized Rickart *-rings, expanding the understanding of their structure through new concepts like generalized right projections and weakly Rickart *-rings.

## Contribution

It defines generalized Rickart *-rings, provides their characterizations, and explores their properties including projections, weak variants, and orthogonal decompositions.

## Key findings

- Every element has a generalized right projection.
- Generalized Rickart *-rings satisfy the parallelogram law.
- Projections in these rings have orthogonal decompositions.

## Abstract

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