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Characterization of \(*\)-(strongly) regular rings in terms of \({\mathcal {G}}\)-projections

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Abstract

A unit-picker is a map \({\mathcal {G}}\) that associates to every ring R a well-defined set \({\mathcal {G}}(R)\) of central units in R which contains \(1_R\) and is invariant under isomorphisms of rings and closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let \({\mathcal {G}}\) be a unit-picker. An element q of a ring R is \({\mathcal {G}}\)-idempotent, a special kind of the strongly regular element, if \(q^{2}=uq\) for some unit picker u of R, or equivalently, \(q=ue\), where u is a unit picker and e is an idempotent of R. In a ring R with involution \(*\), projections are self-adjoint idempotents. As a natural generalization of projections, an element q of a ring R is called a \({\mathcal {G}}\)-projection if \(q^2=uq=uq^{*}\) for some self-adjoint unit-picker u of a \(*\)-ring R, or equivalently, \(q=up\), where p is a projection. We characterize \(*\)-(strongly) regular rings in terms of the \({\mathcal {G}}\)-projection element.

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Acknowledgements

The author is thankful to the referee for his/her valuable comments and suggestions to improve the article.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Art, Erzincan Binali Yıldırım University, 24100, Erzincan, Turkey

    Tufan Özdin

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  1. Tufan Özdin
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Correspondence to Tufan Özdin.

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Communicated by B. Sury.

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Özdin, T. Characterization of \(*\)-(strongly) regular rings in terms of \({\mathcal {G}}\)-projections. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00565-9

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  • DOI: https://doi.org/10.1007/s13226-024-00565-9

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