Abstract
Let R be a unitary ring and a,b ∈ R with ab = 0. We find the 2/3 property of Drazin invertibility: if any two of a, b and a+b are Drazin invertible, then so is the third one. Then, we combine the 2/3 property of Drazin invertibility to characterize the existence of generalized inverses by means of units. As applications, the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism, and the existence and expression of the inverse along a product of two regular elements are obtained, which generalizes the main result of Mary and Patrício (2016) about the group inverse of a product.
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Acknowledgements
We thank Prof. T. Y. Lam for Proposition 2.13 and meaningful suggestions, and thank Prof. D. Khurana for his counterexample which inspires us. We also thank the editor and referees for their constructive comments and suggestions.
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Supported by the National Natural Science Foundation of China (Grant Nos. 12171083, 11871145, 12071070), the Qing Lan Project of Jiangsu Province, and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX22_0231)
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Zhou, Y.K., Chen, J.L. Generalized Inverses and Units in a Unitary Ring. Acta. Math. Sin.-English Ser. 40, 1000–1014 (2024). https://doi.org/10.1007/s10114-023-2196-5
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DOI: https://doi.org/10.1007/s10114-023-2196-5