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.