Rings that are Egyptian with respect to a multiplicative set

Abstract

The notion of an \emph{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 \emph{$W$-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 \emph{not} equivalent to being a distinct sum of unit fractions from $W$, so we need to add the notion of the \emph{strictly} $W$-Egyptian ring. Connections are made with Jacobson radicals, power series, products of rings, factor rings, modular arithmetic, and monoid algebras.

Refereed version. Expanded the introduction and added many little clarifications throughout the paper. 7 pages