Rings of Clean Index 4 and Applications

Abstract

This article is a continuation of the work in [3] on clean index of rings. Motivated by recent work [5] on uniquely clean rings, the clean index of a ring was introduced in [3] to study the structure of rings, where arbitrary rings of clean indices 1, 2, and 3, respectively, are characterized. In particular, clean rings of clean indices 1, 2, and 3 are determined. In this article, we characterize the rings of clean index 4. As applications, clean rings of clean indices 4, 5, 6 and 7 are completely determined.

Key Words:

• Clean index
• Clean ring
• Formal triangular matrix ring
• Local ring
• Morita Context
• Uniquely clean ring

2010 Mathematics Subject Classification:

• 16U99

ACKNOWLEDGMENTS

The authors are grateful to the referee for helpful comments. Part of the work was carried out when the second author was visiting the National Taiwan University sponsored by NCTS of Taipei. He gratefully acknowledges the financial support from NCTS and kind hospitality from the host university. The research of the first author was supported by NSC of Taiwan and NCTS/Taipei, and that of the second author by NSERC of Canada.

Notes

Communicated by T. Albu.