Dedekind Finite Rings and a Theorem of Kaplansky

Abstract

A ring Ris Dedekind Finite(=DF) if xy = 1 implies yx = 1 for all x, yin R. Obviously any subring of a DFring Ris DF. The object of the paper is to generalize, and give a radically new proof of a theorem of Kaplansky on group algebras that are Dedekind finite. We shall prove that all right subrings of right and left self-injective (in fact, continuous) rings are DF.

Dedicated to Ahmad Shamsuddin (1991–2001), in memoriam.

Key Words:

• Dedekind finite
• Self-injective
• Continuous
• Non-singular
• Maximal quotient ring
• Group algebra
• Characteristic 0

Mathematics Subject Classification 2000:

• Primary: 16U60
• Secondary: 16S34, 16D50, 16S90

Acknowledgments

Notes

Dedicated to Ahmad Shamsuddin (1991–2001), in memoriam.