Unifying strongly clean power series rings

ABSTRACT

It is unknown whether a power series ring over a strongly clean ring is, itself, always strongly clean. Although a number of authors have shown that the above statement is true in certain special cases, the problem remains open, in general. In this article, we look at a class of strongly clean rings, which we call the optimally clean rings, over which power series are strongly clean. This condition is motivated by work in [10] and [11]. We explore the properties of optimally clean rings and provide many examples, highlighting the role that this new class of rings plays in investigating the question of strongly clean power series.

KEYWORDS:

• Strongly clean ring
• power series

MATHEMATICS SUBJECT CLASSIFICATION:

• 16U99
• 16W60

Acknowledgments

The authors would like to thank Dr. Warren Wm. McGovern for his time and assistance in completing this paper. The authors would also like to give a hearty thanks to the anonymous referee, whose careful reading and thoughtful suggestions were most appreciated.