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arXiv · 2508.20842

On Generalized Rickart $*$-rings

Anil Khairnar·Sanjay More
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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.

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Anil Khairnar, Sanjay More. 2025-08-28. On Generalized Rickart $*$-rings. https://arxiv.org/abs/2508.20842

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