Jacobson pairs and Bott-Duffin decompositions in rings

@article{Lam2019JacobsonPA,
  title={Jacobson pairs and Bott-Duffin decompositions
 in rings},
  author={T. Y. Lam and Pace P. Nielsen},
  journal={Contemporary Mathematics},
  year={2019}
}
We define the Bott-Duffin decompositions of elements in a ring, which generalize the strongly clean decompositions, and prove that the BottDuffin decompositions of 1− ab are in a natural bijection with those of 1− ba. This bijection respects a number of well known additive decompositions of elements in a ring. For instance, the result implies that 1−ab is strongly clean (respectively, strongly nil-clean, Drazin invertible, quasipolar, or pseudopolar) if and only if so is 1− ba. Examples and… 
Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution
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

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