Finitely presented and coherent ordered modules and rings

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: Ring; module; ordered; finitely presented; finitely related; coherent; system of inequalities; matrix

Domaines: Mathématiques générales [math.GM]

Friedrich Wehrung. Finitely presented and coherent ordered modules and rings. Communications in Algebra, 1999, 27 (12), pp.5893--5919. ⟨10.1080/00927879908826797⟩. ⟨hal-00004049⟩