The aim of this paper is to extend the Bass-Milnor-Serre theorem to the nonstandard rings associated with nonstandard models of Peano arithmetic, in brief to Peano rings.

First, we recall the classical setting. Let k be an algebraïc number field, and let θ be its ring of integers. Let n be an integer ≥ 3, and let G be the group Sln(θ) of (n, n) matrices of determinant 1 with coefficients in θ.

The profinite topology in G is the topology having as fundamental system of open subgroups the subgroups of finite index.

Congruence subgroups of finite index of G are the kernels of the maps Sln(θ) → Sln(θ/I) for which all ideals I of θ are of finite index. By taking these subgroups as a fundamental system of open subgroups, one obtains the congruence topology on G. Every open set for this topology is open in the profinite topology.

We denote by Ḡ (resp., Ĝ) the completion of G for the congruence (resp., profinite) topology.

The Bass-Milnor-Serre theorem [1] consists of the two following statements:

(A) If k admits a real embedding, then we have an exact sequence

That is, Ĝ and Ḡ are isomorphic.

(B) If k is totally imaginary, then one has an exact sequence

where μ(k)is the group of the roots of unity of k.

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take a model to be one of these structures, α, together with some function or relation which associates with every formula A a subset of α. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take a proposition of α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers the primary interpretation associated with the semantics.