## Generalized inverses and their relations with clean decompositions.(English)Zbl 1453.16039

Summary: An element $$a$$ in a ring $$R$$ is called clean if it is the sum of an idempotent $$e$$ and a unit $$u$$ . Such a clean decomposition $$a = e + u$$ is said to be strongly clean if $$e u = u e$$ and special clean if $$a R \cap e R = (0)$$. In this paper, we prove that $$a$$ is Drazin invertible if and only if there exists an idempotent $$e$$ and a unit $$u$$ such that $$a^n = e + u$$ is both a strongly clean decomposition and a special clean decomposition, for some positive integer $$n$$. Also, the existence of the Moore-Penrose and group inverses is related to the existence of certain $$\ast$$-clean decompositions.

### MSC:

 16U90 Generalized inverses (associative rings and algebras) 16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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### References:

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