Valuation domains whose products of free modules are separable

TL;DR

This paper characterizes when the product of free modules over certain valuation domains is separable, linking this property to the maximality of a specific ideal related to the domain's structure.

Contribution

It establishes a precise criterion connecting the separability of module products to the maximality of a particular ideal in valuation domains.

Findings

Product of free modules is separable iff a certain ideal is maximal.

Separable modules are characterized by the maximality condition in valuation domains.

The result applies to valuation domains with countably generated localizations.

Abstract

It is proved that if $R$ is a valuation domain with maximal ideal $P$ and if $R_L$ is countably generated for each prime ideal $L$, then $R^R$ is separable if and only $R_J$ is maximal, where $J=\cap_{n\in\mathbb{N}}P^n$.