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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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