Tadasi Nakayama

On some characteristic properties of quasi-Frobenius and regular rings

Recently the first writer [1] gave a characterization of quasiFrobenius rings, introduced formerly by the second writer [3], in terms of a condition proposed by K. Shoda, which reads: A ring A satisfying minimum condition and possessing a unit element is a quasi-Frobenius ring if and only if A satisfies the following condition :1 (a) every (A-left-) homomorphism of a left-ideal of A into A may be given by the right multiplication of an element of A. In the present note we shall offer a simpler2 proof of this, making use of the second writers former characterization of quasi-Frobenius rings and a theorem in a previous joint note of the writers. The present approach starts, contrary to the one in [1 ],3 with a theorem (Theorem 1) which is independent of any chain condition (or is concerned with maximum condition at most (corollary to Theorem 1) and which is perhaps of interest by itself. We shall also show that a remark at the end of [1], concerning semisimple rings with minimum condition, may be freed from chain condition, to yield a certain characterization of von Neumanns regular rings (Theorem 3).

On Frobenius extensions. II

In Part I we introduced the notion of 2. Frobenius extensions of a ring, as a generalization of Kasch’s [10] Frobenius extensions and hence of classical Frobenius algebras. We proved, in I, bilinear (or sesqui-linear, rather, to follow Bourbaki’s terminology) form and scalar product characterizations of Frobenius extensions in such extended sense, generalizing Kasch’s and classical case, and then studied homological dimensions in them, generalizing and refining the results in Eilenberg-Nakayama [4] and Hirata [6]. Dual bases were considered in case of quasi-free (2.) Frobenius extensions Also the case of a semi-primary or S-ring ground ring was studied.

On an existence lemma in valuation theory

Naturally II includes the infiniteness of prime ideals of 1st degree in an algebraic number field (of finite degree). But the lemmas are perhaps not new. Indeed, II may be proved easily by modifying and generalizing Mor:yas elementary proof to the mentioned particular case. However, since the writers fail to find a literature where these facts are explicitly stated, it is perhaps without use to offer here a proof, indeed a one which is still simpler than, though closely related, the one obtained in the mentioned way. As I follows from II readily, we shall treat II only. Let L-K(a) and let

On the dimension of modules and algebras. V. Dimension of residue rings

We shall consider a semi-primary ring ۸ with radical N (i.e. N is nilpotent and ۸/N is semi-simple (with minimum condition)). All modules considered are left ۸ -modules. We refer to [1] for all notions relevant to homological algebra. The objective of this paper is to establish the following two theorems: Theorem I. Let a be a two-sided ideal in A such that

A remark on Frobenius extensions and endomorphism rings

In his paper [1] F. Kasch developed a theory of Frobenius extensions as a generalization of the theory of Frobenius algebras. In it he established a very interesting relationship between the Frobenius property of an extension and that of its endomorphism ring [1, Satz 5], from which he further derived the Frobenius extension property of Galois extensions of simple rings [1, Satz 6]; with these results he kindly responded to what had been vaguely “conjectured” (as he wrote) by one of the writers on the connection between Galois theory and the theory of Frobenius algebras [2].

A remark on orthogonality relations in finite groups

On the basis of Prof. R. Brauer’s fundamental work, certain orthogonality relations for characters of finite groups have recently been studied by Brauer himself, M. Osima, and one of the present writers; see Iizuka [7] and the references there. In the present short note some general remarks on orthogonality relations, dealing with “blocks” and “sections” of general type, are given first. They are of elementary, and often formal, nature and their proofs are merely combinations of known arguments. So, no deep significance is claimed on them, in comparison with the above alluded results based on deeper arithmetico-group-theoretical considerations. However, applied to blocks and sections of such deeper nature, our remarks give some rather useful informations on them. Thus, for instance, the “maximality” feature of 77-blocks is given a formulation (Prop. 5 below) finer than the one given in [71 Further, some new types of blocks and sections can be constructed, again in application of our remarks to such classical ones. These new blocks and sections give thus new orthogonality relations and we hope that some of them may turn to have some significance. There arize also several problems, which are stated at the end of the present note and to some of which we wish to come back elsewhere.

Construction and characterization of Galois algebras with given Galois group

Recently H. Hasse has given an interesting theory of Galois algebras, which generalizes the well known theory of Kummer fields; an algebra over a field Ω is called a Galois algebra with Galois group G when possesses G as a group of automorphisms and is ( G , Ω )-operator-isomorphic to the group ring G ( Ω ) of G over Ω . On assuming that the characteristic of Ω does not divide the order of G and that absolutely irreducible representations of G lie in Ω , Hasse constructs certain Ω -basis of , called factor basis, in accord with Wedderburn decomposition of the group ring and shows that a characterization of is given by a certain matrix factor system which defines the multiplication between different parts of the factor basis belonging to different characters of G . Now the present work is to free the theory from the restriction on the characteristic. We can indeed embrace the case of non-semisimple modular group ring G ( Ω ).

Note on an ordering theorem for subfields

In a recent paper [3] Tannaka gave an interesting ordering theorem for subfields of a p -adic number field, The purpose of the present note is firstly to observe, on modifying Tannaka’s argument a little, that his restriction to those subfields over which the original field is abelian may be removed and in fact the theorem holds for arbitrary fields which are not p -adic number fields, indeed in a much refined form, and secondly to formulate a similar ordering theorem for algebraic number fields in terms of idele-class groups in place of element groups.