Finitely presented and coherent ordered modules and rings

Abstract

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following features:. (i) Every subring of Q, and every totally ordered division ring, is po-coherent. (ii) For a partially ordered right module Aover a po-coherent poring R Ais po-coherent if and only if Ais a finitely presented .R-module and A +is a finitely generated R +-semimodule. (iii) Every finitely po-presented partially ordered right module over a right po-coherent po-ring is po-coherent. (iv) Every finitely po-presented abelian lattice-ordered group is po-coherent.