Relative $(b, c)$-Inverses with Respect to a Ring Endomorphism
Received:July 06, 2022  Revised:November 25, 2022
Key Words: $\alpha$-$(b, c)$-inverse   Cline's formula   Jacobson's lemma   strongly clean decomposition  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12161049).
Author NameAffiliation
Jun JIAO School of Mathematics and Physics, Anhui University of Technology, Anhui 243032, P. R. China 
Wenxi LI School of Mathematics and Physics, Anhui University of Technology, Anhui 243032, P. R. China 
Liang ZHAO School of Mathematics and Physics, Anhui University of Technology, Anhui 243032, P. R. China 
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Abstract:
      We study the relative properties of $(b, c)$-inverses with respect to a ring endomorphism. A new class of generalized inverses named $\alpha$-$(b, c)$-inverse is introduced and studied in a more general setting. We show by giving an example that $(b, c)$-inverses behave quite differently from $\alpha$-$(b, c)$-inverses. The condition that an $\alpha$-$(b, c)$-invertible element is precisely a $(b, c)$-invertible element is investigated. We also study the strongly clean decompositions for $\alpha$-$(b, c)$-inverses. Some well-known results on ($b, c$)-inverses are extended and unified.
Citation:
DOI:10.3770/j.issn:2095-2651.2023.04.006
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