Decidability for Theories of Modules over Valuation Domains

Decidability of the theory of modules over commutative valuation domains

We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all V -modules is decidable.

Quasi-Primary Decomposition in Modules Over Proufer Domains

In this paper we investigate decompositions of submodules in modules over a Proufer domain into intersections of quasi-primary and classical quasi-primary submodules. In particular, existence and uniqueness of quasi-primary decompositions in modules over a Proufer domain of finite character are proved. Proufer domain; primary submodule; quasi-primary submodule; classical quasi-primary; decompo...

Sigma-cotorsion Modules over Valuation Domains

We give a characterization of Σ-cotorsion modules over valuation domains in terms of descending chain conditions on certain chains of definable subgroups. We prove that pure submodules, direct products and modules elementarily equivalent to a Σ-cotorsion module are again Σ-cotorsion. Moreover, we describe the structure of Σ-cotorsion modules.

Methods for demonstrating consistency and their limitations ( 1932 )

‖ The methods that were used to prove the consistency of formalized theories from the finitist standpoint can be surveyed according to the following classification. ‖1. Method of valuation. It has obtained its essential development by Hilbert’s procedure of trial valuation. Using this procedure Ackermann and v. Neumann demonstrated the consistency of number theory—admittedly, under the restrict...

Divisible Uniserial Modules over Valuation Domains