Some families of strongly clean rings. (English) Zbl 1122.16027

Yang, Xiande; Zhou, Yiqiang

Linear Algebra Appl. 425, No. 1, 119-129 (2007).

A ring \(R\) with identity is called strongly clean if every element of \(R\) is the sum of an idempotent and a unit that commute with each other. For a commutative local ring and for an arbitrary integer \(n\geq 2\), this paper deals with the question whether the strongly clean property of \(\mathbb{M}_n(R[\![x]\!])\), \(\mathbb{M}_n(R[x]/(x^k))\), and \(\mathbb{M}_n(RC_2)\) follows from the strongly clean property of \(\mathbb{M}_n(R)\). This is “yes” if \(n=2\) by a known result.

Reviewer: Tong Wenting (Nanjing)

MSC:

16U60	Units, groups of units (associative rings and algebras)
16S50	Endomorphism rings; matrix rings
16S36	Ordinary and skew polynomial rings and semigroup rings

Keywords:

strongly clean rings; matrix rings; commutative local rings; idempotents; units

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