Abstract In a ring with involution, we prove that a Drazin invertible element is pseudo core invertible if and only if its spectral idempotent is {1, 4}-invertible. As its applications, we obtain necessary and sufficient conditions for 1 − ba (resp., ba) being pseudo core invertible while 1 − ab (resp., ab) has pseudo core inverse, and the pseudo core inverse of 1 − ba (resp., ba) is given in terms of 1 − ab (resp., ab). Inspired by the above idea, Jacobson’s lemma for Moore-Penrose inverse is considered.

中文翻译：

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Jacobson 引理和 Cline 的对合环中的广义逆公式

摘要 在对合环中，我们证明了Drazin可逆元是伪核可逆的当且仅当其谱幂等是{1, 4}-可逆的。作为其应用，我们获得了 1 − ba (resp., ba) 是伪核可逆的充分必要条件，而 1 − ab (resp., ab) 具有伪核逆，而 1 − ba (resp., ab) 的伪核逆., ba) 根据 1 − ab (resp., ab) 给出。受上述思想的启发，考虑了摩尔-彭罗斯逆的雅各布森引理。