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.