2021/04/20 by Ren, Yanxun, Jiang, Lining
#16U99 #FOS: Mathematics #Primary 15A09 #Rings and Algebras (math.RA) #Secondary 16B99
paper · doi:10.48550/arxiv.2104.09846
Let R be an associative ring with unit 1, and a, b, c∈ R satisfy a(ba)2=abaca=acaba=(ac)2a, this paper proves that 1-ac has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively) if and only if 1-ba has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively). In particular, we obtain new common spectral properties for ac and ba in Banach algebras. As applications, new extension of Jacobson's lemma for B-Fredholm elements and generalized Fredholm elements in rings is established.