On ps-Drazin inverses in a ring

Authors : Huanyin CHEN, Tuğçe ÇALCI

Pages : 2114-2124

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :An element $a$ in a ring $R$ has a ps-Drazin inverse if there exists $b\in comm^2insert ignore into journalissuearticles values(a);$ such that $b=bab, insert ignore into journalissuearticles values(a-ab);^k\in Jinsert ignore into journalissuearticles values(R);$ for some $k\in {\Bbb N}$. Elementary properties of ps-Drazin inverses in a ring are investigated here. We prove that $a\in R$ has a ps-Drazin inverse if and only if $a$ has a generalized Drazin inverse and $insert ignore into journalissuearticles values(a-a^2);^k\in Jinsert ignore into journalissuearticles values(R);$ for some $k\in {\Bbb N}$. We show Cline`s formula and Jacobson`s lemma for ps-Drazin inverses. The additive properties of ps-Drazin inverses in a Banach algebra are obtained. Moreover, we completely determine when a $2\times 2$ matrix $A\in M_2insert ignore into journalissuearticles values(R);$ over a local ring $R$ has a ps-Drazin inverse.
Keywords : Generalized Drazin inverse, Cline`s formula, Jacobson`s lemma, 2\times 2 matrix, local ring