Quasi-frobenius (= QF) rings have many interesting characterizations. One such, due to Ikeda [17] is that these rings are right (left) artinian and right (left) self-injective. Thus, if R is QF, then R is right (left) noetherian, so each direct sum of injective right R-modules is injective: in particular, each free, hence, each projective, R-module is injective. One object of this paper is to report that this property characterizes QF-rings:

The fourteenth problem of Hilbert asked finite generation of a certain class of rings and had a counter-example (cf. [4]). On the other hand, many mathematicians gave various sufficient conditions for finite generation of such rings (see, for instance, [9], [5] and [8]). The purpose of the present paper is to give a new sufficient condition. The class of rings to be treated is much more general than those treated before, except for the one in [8].

1. Let F be an algebraic number field of finite degree, and let K/F be a normal extension of degree n. Denote by OK the ring of all integers in K. In we proved the following:

Let E be a totally disconnected compact set in the z-plane and let Ω be its complement with respect to the extended 2-plane. Then Ω is a domain and we can consider a single-valued meromorphic function f(z) in Ω which has a transcendental singularity at each point ζ ∊ E. Suppose that E is a null-set of the class W in the sense of Kametani [4] (= the class NB in the sense of Ahlfors and Beurling [1]).

Many questions about free ideal rings ( = firs, cf. [5] and §2 below) which at present seem difficult become much easier when one restricts attention to local rings. One is then dealing with hereditary local rings, and any such ring is in fact a fir (§2). Our object thus is to describe hereditary local rings. The results on firs in [5] show that such a ring must be a unique factorization domain; in §3 we prove that it must also be rigid (cf. the definition in [3] and §3 below). More precisely, for a semifir R with prime factorization rigidity is necessary and sufficient for R to be a local ring.

A subgroup was defined by O. Ore to be quasinormal in a group if it permuted with all subgroups of the group, and he proved [5] that such a subgroup is subnormal (= subinvariant = accessible) in a finite group. Finite groups in which all subgroups are quasinormal were classified by K. Iwasawa [3], and more recently N. Ito and J. Szép [2] and the author [1] proved that a quasi-normal subgroup is an extension of a normal subgroup by a nilpotent group. Similar results were obtained by O. Kegel [4] and in [1] for subgroups which permute not necessarily with all subgroups but with those having some special property.

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

As an extension of a result due to W. D. Barcus and J. P. Meyer [4], T. Ganea [8] has recently proved a theorem concerning the fibre of the extension E∪CF→B of a fibre map p: E→B to the cone CF erected over the fibre F. In this paper we shall establish a generalized Ganea theorem which asserts that the homotopy type of the fibre of a canonical extension ξ′ of a triad (cf. [13]) is determined by those of f and g (see Theorem 3.4).

Let G be a connected semi-simple algebraic group defined over Q and let Γ be a discrete subgroup of GR (the subgroup of G consisting of points rational over R) such that Γ\GR is compact. The main purpose of the present paper is to prove that for a certain type of group G there exists an invariant algebraic differential from ω on G of highest degree defined over Q such that

A field k is called quasi-finite if it is perfect and if Gk≈Ż where Gk is the Galois group of the algebraic closure kc over k and Ż is the completion of the additive group of the rational integers. The classical reciprocity law on the local field with finite residue field is well-known to hold on local fields with quasi-finite residue field ([4] [5]). Thus it is natural to ask if the global reciprocity law should hold in the ordinary sense (see § 1 below) on the function-fields of one variable over quasi-finite field. We consider here two basic prototypes of non-finite quasi-finite fields:

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.