Which Finite Rings Have the Most or Fewest Units?

Abstract

Among finite rings of a fixed order that are not fields, we classify those that have the most units. Our methods also give the finite rings that have the fewest units.

MSC::

• Primary 16U60
• Secondary 16P10

Acknowledgments

We are grateful to two anonymous reviewers for comments and recommendations which have made for a more readable article. Our presentation has also benefited from Susan Colley’s eagle editorial eye.

Additional information

Notes on contributors

Jonathan Cohen

JONATHAN COHEN received his doctorate from the University of Maryland. He was a postdoctoral visitor at the University of Oklahoma during the writing of this article but will soon make a short move down I-35 to the University of North Texas. His primary research interests are in number theory and p-adic representation theory.

Department of Mathematics, University of Oklahoma, Norman, OK 73019

[email protected]

Alan Roche

ALAN ROCHE studied at University College Dublin and the University of Chicago, and now teaches at the University of Oklahoma. His primary mathematical interests lie in number theory and representation theory. He is an ailurophile.

Department of Mathematics, University of Mathematics, Norman, OK 73019

[email protected]