Segregated algebras have been nicely characterized by M. Ikeda [4]. In this paper §1, we consider segregated rings and study the structure of such rings in Theorems 1.1 and 1.2. In §2, we specialize to the case of segregated algebras of finite dimension over a field. Theorem 2.1 gives a new characterization of such algebras. Theorem 2.2 shows an interesting property of segregated algebras; two segregated algebras S and T, with radicals N and P respectively, are isomorphic if and only if S/N2 and T/P2 are isomorphic.

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

Let K be a function field of one variable over a perfect field k and let v be a valuation of K over k. Then is the different-divisor (Verzweigungsdivisor) of K/k(x), and (x)∞ is the denominator-divisor (Nennerdivisor) of x. In §1 we consider a generalization of this theorem in the function fields of many variables under some conditions. In §2 and §3 we consider the differential forms of the first kind on algebraic varieties, or the differential forms which are finite at every simple point of normal varieties and subadjoint hypersurfaces which are developed by Clebsch and Picard in the classical case. In §4 we give a proof of the following theorem. Let Vr be a normal projective variety defined over a field k of characteristic 0, and let ω1, …, ωs be linearly independent simple closed differential forms which are finite at every simple point of Vr. Then the induced forms on a generic hyperplane section are also linearly independent.

J. L. Doob, D. Blackwell, W. Feller and other authors have obtained several results concerning the renewal theorem. Especially Doob [1] has considered the renewal process and has showed that it becomes a stationary Markov process if we add a certain initial random variable to it. In the present note, we shall study this stationary Markov process and try to determine its transition probability by virtue of a pair of partial differential equations.

The author would like to express his hearty thanks to prof. A. Amakusa who has encouraged him with kind discussions throughout the course of preparing the present note.

La recherche de la structure des espaces homogènes kählériens d’un groupe de Lie semi-simple fait l’objet de plusieurs travaux récents. L’étude détaillée a été faite au cas compact. Il a été montré en particulier que ces espaces homogènes compacts sont des variétés algébriques (même rationnelles) et simplement connexes.

Le but essentiel de ce travail est de montrer que tout espace homogène kählérien compact est produit direct d’un tore complexe et d’un espace homogène kählérien d’un groupe de Lie compact semi-simple (Théorème 3). Pour ce but nous étudierons au paragraphe 1 la structure des espaces homogènes symplectiques d’un groupe de Lie réductif. La structure et la situation du groupe d’isotropie seront clarifiées dans le théorème 1. Au paragraphe 2 on en déduit un théorème sur la structure des espaces homogènes kählériens d’un groupe de Lie réductif (Théorème 2). Le théorème 3 est une conséquence immédiate du théorème 2.

Throughout this paper all rings are assumed to have unit elements. A ring Λ is said to be semi-primary if its Jacobson radical N is nilpotent and Г = Λ/N satisfies the minimum condition. The main objective of this paper is

THEOREM I. Let A be a semi-primary algebra over a field K. Let N be the radical of Λ and Г = Λ/N. If

Then

dim Λ = gl.dim Λ.

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) <∞ and if the cohomolgical dimension of Ω, dim Ω, is ≦ 1, then every residue-algebra of Ω has a finite cohomological dimension. In the present note we prove a theorem of converse type, which gives, when combined with the cited result, a rather complete general picture of algebras whose residue-algebras are all of finite cohomological dimension.

The present paper is devoted to the study of differential geometry of Kaehlerian homogeneous spaces. In section 1 we deal with the canonical decomposition of a simply connected complete Kaehlerian space and that of its largest connected group of automorphisms. We know that a simply connected complete Riemannian space V is the product of Riemannian spaces V0, V1, …, Vn, where V0 is a Euclidean space and V1, …,Vn are not locally flat and their homogeneous holonomy groups are irreducible [2]. Moreover, if V is homogeneous, so are all Vk [10]. We shall show that if V is Kaehlerian space with real analytic metric (resp. Kaehlerian homogeneous space), each factor Vk is also Kaehlerian (resp. Kaehlerian homogeneous) and that V is the product of V0, V1, …, Vn as Kaehlerian space. We call this decomposition the de Rham decomposition of the Kaehlerian space V. Although this result is supposedly known, there is no published proof as yet. Using this decomposition theorem we shall show that the largest connected group of automorphisms of a simply connected complete Kaehlerian space with real analytic metric is the direct product of those of the factors of the de Rham decomposition. In the Riemannian case this result has be been established in [3] by one of the authors of the present paper.

In the previous paper [7], we have studied the relationship between the Riemannian connection of an n-dimensional Riemannian space M imbedded into the (n + k)-dimensional Euclidean space Rn+k and the canonical connection in the bundle Pn, k = O(n + k)/{l} Х O(k) over the Grassmann manifold Mn, k = O(n + k)/O(n) Х O(k).

In the first half of the present paper, the relationship between the canonical connections in bundles and the invariant Riemannian connection on Mn, k will be discussed. We obtain the holonomy groups of these canonical connections.

We have proved in [2] that if the restricted homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at every point, then the Riemannian manifold is locally symmetric, that is, the covariant derivatives of the curvature tensor field are zero. The proof of this theorem, however, depended on an insufficiently stated proposition (Theorem 1, [2]). In the present note, we shall give a proof of a more general theorem of the same type.

Is every isometry, or more generally, every affine transformation of a Kählerian manifold a complex analytic transformation? The answer is certainly negative in the case of a complex Euclidean space. This question has been recently studied by Lichnerowicz [8] and Schouten-Yano [11] from the infinitesimal point of view; they have found some conditions in order that every infinitesimal motion of a Kählerian manifold preserve the complex structure. (As a matter of fact, [11] has dealt with the case of a pseudo-Kählerian manifold, which does not differ essentially from a Kählerian manifold as far as the question at hand is concerned.)

In the present paper, we generalize their results by a different approach. In order to explain our main idea, we shall first give a few definitions (1 and 2) and state our main results (3). The proofs are given in the subsequent sections.

Let K be any field, and L a separable extension of K of finite degree. L has a structure of vector space over K, and we shall denote this space by V. The space of endomorphisms of V will be denoted by Let x be any element of L, and N(x) the norm of x relative to the extension L/K. N is then a function defined on V with values in K. We shall call N the norm form on V. The multiplicative groups of non-zero elements of K and L will be denoted by K* and L* respectively. Let H be any subgroup of if K*. Then the elements z of L* such that N(z)∈H form a subgroup of L*, which we shall denote by GH. On the other hand the elements s of such that N(sx) = Λ(s)N(x) with Λ(s)∈H for all X∈V, form obviously a subgroup of GL(V), which we shall denote by becomes an algebraic group if H=K* or {1}. In case will mean the group of linear transformations of V leaving semi-invariant the norm form of L/K and in case will mean the group of linear transformations of V leaving invariant the norm form of L/K.

La question posée par Noshiro dans [5] pour les ensembles d’accumulation relatifs à des fonctions pseudo-analytiques classiques a été résolue par (T.) Yosida [12] premièrement et ensuite, le présent auteur [8] a étendu son résultat (théorème 1 de [12]) au cas où un nombre fini ou dénombrable de surfaces de Riemann, qui sont des surfaces de recouvrement, sont transformées dans une autre surface de Riemann par une application continue à dérivées partielles continues sauf en un ensemble fermé de mesure linéaire nulle, ayant pour image un ensemble de mesure linéaire également nulle; le quotient de dilatation n’y est pas nécessairement borné mais subit certaine condition, et les ensembles d’accumulation y sont définis pour les éléments frontières de Kerékjartô-Stoïlow. Une généralisation d’un théorème étoilé de Gross obtenue dans [7] a été utilisée dans sa démonstration. L’auteur a continué son étude sur ce théorème étoilé et un de ses résultats établis dans [9] lui permettra d’améliorer les résultats de [8] dans le présent mémoire. L’idée fondamentale est la même que dans [8] mais nous la récrirons, en nous reportant à [8] le moins possible.

Soit V une variété complexe et soit I le champ de tenseurs de type (1,1) de V définissant la structure complexe de V. Un champ de vecteurs ξ sur V sera dit conforme si I [ξ, η]= [ξ, Iη] pour tout champ de vecteurs η sur V. On désignera par α l’ensemble de tous les champs de vecteurs comformes sur V. α est une sous-algèbre de Lie de l’algèbre de Lie de tous les champs de vecteurs sur V. Si V est compacte, α est de dimension finie et s’identifie avec l’algèbre de Lie du groupe de Lie A(V) d’homéomorphismes analytiques de V [2] De plus, si ξ, η ∈ α, on a On peut donc définir une structure d’algèbre de Lie complexe de α en posant pour tout ξ ∈ α.

D’abord nous définissons les ensembles de Cantor généralisés. Soient k1, k2, … des nombres entiers supérieurs à 1 et soient p1, p2, … des nombres finis quelconques également supérieurs à 1. On pose lq=l/(kqpq). Soit I un intervalle de longueur d > 0. On enlève de I (kq — l) intervalles de même longueur tels qu’il reste kq intervalles de même longueur lqd. On appelle cette opération la q-opération appliquée à I. On commence par appliquer l’1-opération à [0,1], on applique la 2-opération à chacun des intervalles qui restent, puis on applique la 3-opération à chacun des intervalles qui restent, et ainsi de suite. On appellera l’ensemble limite restant un ensemble de Cantor généralisé dans E1 et le notera F = F(kq, pq). Notre définition dans En = {(x1, …, xn)}, l’espace euclidien à n dimensions (n ≧ 2), est la suivante: Soit un ensemble de Cantor généralisé défini sur l’axe de xj; nous appellerons l’ensemble produit F = F1 × … × Fn un ensemble de Cantor généralisé dans En (n ≧ 2). II sera appelé symétrique si F1 = … =Fn.