2005/01/25 by Friedrich Wehrung
Mathematics · #math.GM #msc:06F25 #msc:16W80 #msc:12J15 #msc:15A39 #msc:08C15
published as Communications in Algebra 27, no. 12 (1999) 5893--5919
arxiv created 2005/01/25 · arxiv updated 2009/12/01
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 A over a po-coherent poring R, A is po-coherent if and only if A is 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 presented abelian lattice-ordered group is po-coherent.