On strongly clean matrix and triangular matrix rings. (English) Zbl 1114.16024

Chen, Jianlong; Yang, Xiande; Zhou, Yiqiang

Commun. Algebra 34, No. 10, 3659-3674 (2006).

A ring \(R\) is called strongly clean if every element of \(R\) is the sum of an idempotent and a unit that commute. In this paper the authors obtain new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proved that the \(2\times 2\) matrix ring over the ring of \(p\)-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean. For a commutative ring \(R\), the authors prove that the \(n\times n\) upper triangular matrix ring over \(R\) is strongly clean for every \(n\).

Reviewer: Tong Wenting (Nanjing)

MSC:

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

Keywords:

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

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