On Strongly Clean Matrix and Triangular Matrix Rings

Abstract

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.

Key Words:

• Idempotent
• Local ring
• Matrix ring
• Strongly clean ring
• Triangular matrix ring
• Unit

Mathematics Subject Classification:

• Primary 16U99, 16S50, 16S99
• Secondary 16U60, 16U10

ACKNOWLEDGMENTS

The research was carried out during a visit by the first author to Memorial University of Newfoundland. He would like to gratefully acknowledge the financial support and kind hospitality from his host institute. The first author was supported by NNSF of China (No.10171011), the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutes of MOE (China), and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education (China). The third author is supported by NSERC (Grant OGP0194196) and a grant from the Office of Dean of Science, Memorial University.

Notes

Communicated by R. Wisbauer.