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Borooah, Gautam; Diesl, Alexander J.; Dorsey, Thomas J.

Strongly clean matrix rings over commutative local rings. (English) Zbl 1162.16016

J. Pure Appl. Algebra 212, No. 1, 281-296 (2008).
Throughout \(R\) is an associative ring with identity. A ring is called strongly clean if every element is the sum of an idempotent and a unit that commute with each other.
The authors completely characterize the commutative local rings \(R\) for which the matrix ring \(\mathbb{M}_n(R)\) is strongly clean, in terms of factorization in \(R[t]\). Among several other results, it is shown that for any monic polynomial \(f\in R[t]\) , the strong cleanness of the companion matrix of \(f\) is equivalent to the strong cleanness of all matrices with characteristic polynomial \(f\). Various interesting examples are given and open questions are raised.
Reviewer: Iuliu Crivei (Cluj-Napoca)

Cited in 2 Reviews
Cited in 27 Documents

MSC:

16S50 Endomorphism rings; matrix rings
16U60 Units, groups of units (associative rings and algebras)
13H99 Local rings and semilocal rings

Keywords:

strongly clean rings; strongly \(\pi\)-regular rings; commutative local rings; matrix rings; idempotents; strongly clean elments; companion matrices
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\textit{G. Borooah} et al., J. Pure Appl. Algebra 212, No. 1, 281--296 (2008; Zbl 1162.16016)
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