Dedekind-Finite Strongly Clean Rings

Abstract

In this article we partially answer two open questions concerning clean rings. First, we demonstrate that if a quasi-continuous module is strongly clean then it is Dedekind-finite. Second, we prove a partial converse. We also prove that all clean decompositions on submodules of continuous modules extend to the entire module.

Key Words:

• Dedekind-finite
• Strongly clean ring

2010 Mathematics Subject Classification:

• Primary 16E50
• Secondary 16D50, 16D70
• 16U99

ACKNOWLEDGMENT

We thank T. Y. Lam and the Berkeley ring theory group for useful conversations concerning capably clean rings. Much of the material which appears in the second section arose from these discussions. The third author was partially supported by the University of Iowa Department of Mathematics NSF VIGRE grant DMS-0602242.

Notes

Communicated by J. L. Gomez Pardo.