Topological Ordered Modules

Abstract

Sufficient conditions are found for the order topology on an ordered module over an ordered ring to be a module topology.More concretely, if $M$ is an ordered module over a topological unital ordered ring such that $M^+\neq \{0\}$, $M^+\backslash\{0\}$ is downward directed, $0\in \cl\left(M^+\backslash\{0\}\right)$, and the module ordering is strong, then the order topology is a module topology. Nontrivial examples are provided showing that these sufficient conditions are not necessary.