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As a generalization of the notion of Frobenius algebras over a field Kasch [103 introduced that of Frobenius extensions of a ring. The present writers [13] recently freed one of Kasch’s main theorems from its rather strong S-ring assumption of the ground ring. However, even with the removal of the S-ring assumption of the ground ring the notion does not seem general enough, and we wish, in the present paper and its sequel, to develope the theory upon the basis of a more general notion of Frobenius extensions. Thus, we replace the free module property of the extension by the projective module property (according to a general tendency in algebra), which has been done in fact in case of Frobenius algebras over a commutative ring in a previous work by Eilenberg and one of the writers [4], and, further, take automorphisms of the ground ring into the definition of Frobenius extensions (which seems quite natural particularly in case of non-commutative rings). To such generalized notion of Frobenius extensions we may extend many of Kasch’s theorems, including those which are immediate extensions of classical theorems for Frobenius algebras and those which are essentially new, as the above alluded endomorphism ring theorem. Also homological properties of Frobenius extensions, as were developed in Hirata’s [6] recent paper in succession to Eilenberg-Nakayama [4], can be extended to our present generalized case; we shall also exceed [4], [6] somewhat in considering injective and weak dimensions.

The purpose of the present work is to give, as a continuation of the writer's study of Galois theory for general rings ([8], [9], [10]), a kind of Galois theory for general, non-commutative and non-semisimple rings, which includes, at least in its main features, the Kaloujnine-Jacobson Galois theory of non-normal fields ([3]; cf. [4], [5]). To deal with the non-commutativity we bring to the fore certain double-moduli rather than self-composites, while the non-semisimplicity is manipulated by the method and idea used in the writer's above mentioned study on (normal) Galois theory and commuter systems of nonsemisimple rings. (For the normal Galois theory of rings cf. [1], [2], [6], [7], [11], besides the above.) Some of our arguments may even serve to make some simplification in Jacobson's treatment of ordinary fields.

A ring (with unit element) Λ is called semi-primary if it contains a nilpotent two-sided ideal N such that the residue ring Γ = Λ/N is semi-simple (i.e. l.gl.dim Γ = r.gl.dim Γ = 0). N is then the (Jacobson) radical of Λ. Auslander [1] has shown that if Λ is semi-primary thenThe common value is denoted by gl. dim Λ. On the other hand, for any ring Λ the following conditions are equivalent : (a) 1. gl. dim Λ ≦ 1, (b) each left ideal in Λ is projective, (c) every submodule of a projective left Λ-module is projective. Rings satisfying conditions (a)-(c) are called hereditary. For integral domains the notions of “hereditary ring” and “Dedekind ring” coincide.

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.

Let K/k be a Galois extension. Formerly the writer studied, in [3], [4], a certain correlation of factor sets in K/k with the norm class group of K/k, and extended it, in [5.], to 3-dimensional cocycles. The present note is to study the same relationship for general n-cocycles. As a matter of fact, the constructions which underlie the relationship have become common places in cohomology theory, through the works of French and American authors, and indeed the construction to bring certain (non-Galois) cocycles into the ground field k has been discussed by Baer [1, Theorem C] for general dimensions n under the setting of general group cohomology.

Recently M. Ikeda [1] succeeded in determining the structure of absolutely segregated algebras, i.e. algebras whose 2-cohomology groups all vanish. His beautiful result reads : an algebra A, of finite rank over its ground field, is absolutely segregated if and only if i) the residue-algebra A/N modulo the radical N is separable and, moreover, ii) the A-left-module N is an (Mo)-module. A. simplification was given by H. Nagao [5], who obtained, besides an interesting-result on algebras with vanishing 3-(or higher) cohomology groups, an elegant short proof to the fact that under the assumption of i), the property ii) is necessary, and sufficient, for the absolute segregation of A.

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 writer's 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 Neumann's regular rings (Theorem 3).

In this paper we prove the fundamental exact sequence in (co)homology for non-normal subgroups announced in our previous note [8]: Let H be a subgroup of a group G. If M is a G-module and if, for a natural number n, Hm(U, M) = 0 for m = 1, …, n - 1 and for every subgroup U of H which is an intersection of conjugates of H in G, then we have an exact sequencethe significance of the maps and the group Hn(H, M, M)1 will be explained below. (We have a dual result for cohomology groups).

In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension ≦n In a subsequent paper [4] Eilenberg gave an alternative treatment of the same question. The present paper is devoted to the discussion of a number of questions suggested by the results of [4] and [5]. Among others it is shown that the conditions employed in stating the main results in [4] and [5] are equivalent, so that the main results of these two papers are in accord. Further, the cohomological dimension of a residue-algebra is studied in terms of that of the original algebra and the (module-) dimension of the associated ideal. The terminology and notation employed here are that of [3].

Let F be a commutative ring with unit element 1. A ring R is called an algebraic ring over F by Drazin, in his recent paper [2], when R is an algebra over F and every element x of R satisfies an equation of lower monic form

Introduction. The outer Galois theory, started by Jacobson [8], has been developed rather thoroughly [2; 3; 6; 12; 16]. The general Galois theory, dealing with general groups of automorphisms (with some restrictions though), has been established by Cartan [51 and Jacobson [9] in case of sfields. The purpose of the present paper is to offer a similar theory for simple rings with minimum condition('). The same has been given in fact in Hochschild [7] for simple algebras (finite over their centers). But the method breaks down in the case of general simple rings, infinite over their centers, and a new approach is necessary(2). The writer [14] has recently shown that if A is a simple ring and C is a weakly normal (cf. ?1 below) simple subring of A, then the A-left-, C-right-module A is fully reducible, and he has applied this fact, together with some methods in Dieudonne [6], to obtain a theorem of extension for isomorphisms in simple rings. It turns out that this full reducibility of A, with respect to the left-multiplication of A and the rightmultiplication of C, and some crossed product theorems, proved and used in the older papers by Azumaya and the writer [3; 12; 13; 16], are appropriate means for establishing the Galois theory(') for simple rings. In fact, if A is in particular a sfield, then A is clearly minimal (=irreducible) with respect to any operator domain containing the left-multiplication ring of A, and this fact underlies the Galois theory, as well as many other theories, for sfields. It is replaced, when A is a simple ring, by our A-C-full reducibility. The first section of the present paper gives some preliminary, though fundamental, lemmas on weakly normal simple subrings of a simple ring. In ?2 we introduce regular groups of automorphisms of a simple ring, which are the class of automorphism groups employed in our Galois theory, and consider their invariant systems. Conversely, we consider in ?3 the group of automorphisms leaving a subring, of a certain type, elementwise fixed. The Galois theory follows then in ?4. Although our method is rather different, we follow there the pattern of the algebra case in Hochschild [7 ]. Our theory can

Let G be a finite group. A(left) G-module A of G is said to be of trivial cohomology when Hn(H, A)= 0 for all rational integers n and for all subgroups H of G. The main purpose of the present note is to determine the structure of finitely generated G-modules of trivial cohomology, which turns out to be remarkably simple (See Theorem 1 and Corollary 3 below). We prove also an (easy) localization theorem for cohomological triviality.

It was shown in Eilenberg-Nagao-Nakayama [3] (Theorem 8 and § 4) that if Ω is an algebra (with unit element) over a field K with (Ω: K)

The duality theorem for inoncommutative compact groups was established by Tannaka,1 and later, but independently, by Krein.2 Different proofs have been given by Bochner and Yosida.3 The purpose of the present note is to supplement the theorem by a duality between (closed) subgroups and certain representation-theoretically defined systems. It provides also an extension of Tannaka's theorem itself. As a matter of fact, it is more or less known that the subgroups are in 1-1 correspondence with, say, left-invariant subrings with conjugation of the ring of coefficients of (irreducible) representations.4 But we propose to make the dual objects to be associated with subgroups free from the properties concerned with the group operation, such as left-translations, and obtain purely representationtheoretical dual systems.5 Our intention is also to have a formulation which gives directly the subgroups of a character group in the abelian case. Thus our work has no novelty in analytical respects, and our concern lies mainly in its formulation. 1. Let G be a topological group. We take a representation

Formerly Hua [1] proved that if A is a division ring with centre Z and if there exists a natural number n such that an ∈ Z for every a ∈ A, then A is commutative; this generalizes Wedderburn's theorem on finite division rings. Another generalization of Wedderburn's theorem, due to Jacobson [3], asserts that every algebraic division algebra over a finite field is commutative. On the other hand, a theorem of Noether and Jacboson [3] states that every noncommutative algebraic division algebra contains an element which is not contained in the centre Z and is separable over Z.

Recently one of the writers used, in proving a theorem on the commutativity of certain division rings, the following lemma :I. Let L be a field and K be its proper subfield. Except either when L is of characteristic p ≠ O and absolutely algebraic or ivhen L is algebraic and purely inseparable over K, there exists a pair of distinct (special exponential) valuations in L which coincide on K.II. Let K be a field which is either of characteristic 0 or not absolutely algebraic, and L be its separable finite extension. There exist then infinitely many valutions in L which are of 1st degree over K.

A conjecture concerning characterization of blocks of representations of symmetric groups by means of Young diagrams, given formerly by one of the writers, was affirmed recently by R. Brauer and G. de B. Robinson jointly. Namely, it was proved in Brauer’s paper, relying on his general, and profound, theory of blocks in finite groups, that the number of p-blocks in a symmetric group of degree n, is exactly equal to the number of p-cores, or p-kernels in the terminology of NI, with n- Ip (l= 0, 1, 2,…) nodes, and that certain statements designated as (A), (B) and (C) would secure that two irreducible representations with a same p-core belong to a same block (whence the conjecture itself when combined with the estimate of the number of blocks). Then these statements (A), (B), (C) were proved actually in Robinson’s paper.

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).

The following small remark on the relationship of the (quasi-)regularity and the zero-divisor property may be more or less known but does not seem to the writer to have been explicitly stated in a literature.