On two open problems about strongly clean rings

By Zhou Wang, Jianlong Chen. Bulletin of the Australian Mathematical Society, 2004

47 citations recorded by the source corpus.

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. In 1999 Nicholson asked whether every semiperfect ring is strongly clean and whether the matrix ring of a strongly clean ring is strongly clean. In this paper, we prove that if R = {m/n ∈ ℚ: n is odd}, then M2(R) is a semiperfect ring but not strongly clean. Thus, we give negative answers to both questions. It is also proved that every upper triangular matrix ring over the ring R is strongly clean.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dd9aa70741d3c7e531a41a74164d23925be61ec5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. A. Diesl
2. Gautam Borooah
3. Thomas J. Dorsey
4. Yiqiang Zhou
5. Jianlong Chen
6. Huanyin Chen
7. Peter Danchev

In Geometry and Topology

1. Yiqiang Zhou
2. A. Diesl
3. Gautam Borooah
4. Thomas J. Dorsey
5. Jianlong Chen
6. Huanyin Chen

Citations in the corpus

• On Strongly Clean Matrix and Triangular Matrix Rings (2006, background): “This fact was recently observed by Wang and Chen ( 2004 ) and by Sanchez Campos (Unpublished) independently (answering two questions of Nicholson, 1999 ).”
• Strongly clean triangular matrix rings over local rings (2007, background): “[19] Let R be a uniquely clean commutative local ring.”
• Strongly clean matrix rings over commutative local rings (2008, background): “Our results will place the example from [25] that M2(Z(2)) is not strongly clean in a broader context, and will, in particular, provide a conceptual proof of this result.”