## Strongly clean rings and Fitting’s lemma.(English)Zbl 0946.16007

The author defines a ring $$R$$ to be strongly clean if every element of $$R$$ is a sum of an idempotent and a unit which commute. He gives an elementary proof of the fact that every strongly $$\pi$$-regular ring is strongly clean, which was originally proved by W. D. Burgess and P. Menal using Pierce sheaves [Commun. Algebra 16, No. 8, 1701-1725 (1988; Zbl 0655.16006)]. Strong $$\pi$$-regularity in the endomorphism ring of a module $$M$$ is known to be equivalent to $$M$$ satisfying Fitting’s Lemma, which can be stated as saying that for any $$\alpha\in\text{End}(M)$$, there is an $$\alpha$$-invariant decomposition $$M=P\oplus Q$$ such that $$\alpha|_P$$ is an automorphism of $$P$$ while $$\alpha|_Q$$ is nilpotent [E. P. Armendariz, J. W. Fisher, and R. L. Snider, Commun. Algebra 6, 659-672 (1978; Zbl 0383.16014)]. Here the following analog is proved: $$\text{End}(M)$$ is strongly clean if and only if for any $$\alpha\in\text{End}(M)$$, there is an $$\alpha$$-invariant decomposition $$M=P\oplus Q$$ such that $$\alpha|_P$$ and $$(1-\alpha)|_Q$$ are automorphisms of $$P$$ and $$Q$$, respectively. Many of the statements quoted above are actually established for individual ring elements or endomorphisms.

### MSC:

 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16U60 Units, groups of units (associative rings and algebras) 16S50 Endomorphism rings; matrix rings 16W20 Automorphisms and endomorphisms

### Citations:

Zbl 0655.16006; Zbl 0383.16014
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### References:

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