In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

The main purpose of this note is to give a characterization of p-pure unital subrings of the p-adic completion of the ring of integers R of an algebraic number field K localized at a maximal ideal p. This yields a characterization of the valued subfields of the p-adic field. In this context there turn up valuations of rational function fields in many indeterminates which seem to be new. The proof that the underlying function is indeed a valuation is quite easy here, however direct computations would involve a large amount of combinatorics. Our approach seems to fit well with Kronecker's, apparently forgotten, approach to ideal theory in rings of algebraic integers [3]. The concept of p-pure unital subrings arose from a study by the first author and A. Laroche of quasi-p-pure-injective (q.p.p.i.) abelian groups ([1], p. 582).

Let t be a field and a finite dimensional t-algebra. Auslander-Reiten sequences [AR] play a fundamental rôle in the representation theory of ; in particular, they can be used to construct new indecomposable modules from known ones. For the latter reason I think it worthwile to point out certain torsion theories on the category of -modules, such that the category of -torsionfree modules has Auslander-Reiten sequences; thus giving another construction of indecomposable modules.

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall prove

THEOREM. If RG ≌ RH as R-algebras, then G ≌ H

The question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.

In proving the above result, it is necessary to study closely the category of extensions (ℊs , S), where the objects are short exact sequences of SG-modules