Rings that are Egyptian with respect to a multiplicative set

Abstract

The notion of an Egyptian integral domain D (where every fraction can be written as a sum of unit fractions with denominators from D) is extended here to the notion that a ring R is W-Egyptian, with W a multiplicative set in R. The new notion allows denominators just from W. It is shown that several results about Egyptian domains can be extended to the W-Egyptian context, though some cannot, as shown in counterexamples. In particular, being a sum of unit fractions from W is not equivalent to being a distinct sum of unit fractions from W, so we need to add the notion of the strictly W-Egyptian ring. Connections are made with Jacobson radicals, power series, products of rings, factor rings, modular arithmetic, and monoid algebras.

KEYWORDS:

• W-Egyptian
• denominators
• Egyptian integral domain D
• Jacobson radicals
• modular arithmetic
• monoid algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13F99
• Secondary: 13B30

Acknowledgments

I offer here my thanks to the anonymous referee, who insisted that I provide explanations herein to make the paper more self-contained. As a result, I believe the paper reads better now. The referee also gave several useful corrections and suggestions which also have improved the paper.