The PRIMITIVE LOGIC LO introduced in my former work is a logic having only two logical constants IMPLICATION → and UNIVERSAL QUANTIFICATION ( ) with their usual inference rules which are admitted even in the INTUITIONISTIC PREDICATE LOGIC LJ. LO is really a very simple logic, maybe the simplest possible logic as one can imagine, but it is very important because of its universal character. In fact, popular logics such as the LOWER CLASSICAL PREDICATE LOGIC LK, the INTUITIONISTIC PREDICATE LOGIC LJ, the MINIMAL PREDICATE .LOGIC LM, etc. can be faithfully interpreted in it. Speaking frankly, I am further expecting that all the important logics would be interpreted faithfully in it and would disclose their intrinsic characteristics by being interpreted in it. Main purpose of this paper is to show the universal character of the primitive logic LO by pointing out that a series of typical logics are faithfully interpretable in LO.

The concept of regular sequence of elements of a ring A (first introduced by Serre under the name of A-sequence [2]), has far-reaching uses in the theory of local rings and in algebraic geometry. It seems, however, that it loses much of its importance when A is not a noetherian ring, and in that case, it probably should be superseded by the concept of quasi-regular sequence [1].

The main objective of this paper is to study the action of the group of differentiate structures Γ2n-1 on the (2n-1)-sphere S2n-1 on the diffeomorphism classes on the real projective (2n-1)-space P2n-1 by connected sum. This is done by considering universal covering spaces of the connected sum where Σ is an exotic (2n-1)-sphere.

Bourbaki has established the following theorem which we state without proof as

THEOREM A ([3, Theorem 6, §4]). Let R be a noetherian integrally closed domain and M a finitely generated torsion free R-module. Then there exists a free submodule F of M such that M/F is isomorphic to an ideal in R.

It is our purpose in this note to present a few consequences of this theorem. Before giving these results we briefly review some terminology and known results.

In der Arbeit [2] hat L. Rédei den Begriff des einstufig nichtkommutativen Ringes eingefiihrt, und die alien möglichen Typen der einstufig nichtkommutativen endlichen Ringe aufgezàhlt. Sie bestehen aus drei Grundtypen und ihren homomorphen Bildern. In der oben genannten Arbeit hat Rédei einstufig nichtkommutativen unendlichen Ringes ganz kurz erwähnt, und nur darauf hingewiesen, dass man nicht einmal weiss, ob solche Ringe überhaupt existieren. In der vorliegenden Arbeit handelt es sich um die Nichtexistenz der einstufig nichtkommutativen unendlichen Ringe. Obwohl die Nichtexistenz der allge-meinen unendlichen Ringe vom genannten Charakter hoch glaubwürdig scheint, kann ich doch zur Zeit die Aufgabe in ihrer vollen Allgemeinheit nicht losen, sondem ich kann nur zeigen, dass ein einstufig nichtkommutativer Ring mit einer zusätzlichen Bedingung notwendigerweise endlich wird. Im folgenden soil nämlich der untenbenannte Satz bewiesen werden.