In a previous paper, we studied a general theory of integrally closed Henselian integrity domains and some properties of Henselian valuation rings. The present paper is its continuation. The main aim of the present paper is to study a general theory of Henselian local integrity domains in the present paper we call a ring o a local ring if o is a quasi-local ring and if the intersection of all powers of the maximal ideal of o is zero, and in this case we introduce a topology by taking the system of all powers of the maximal ideal as a system of neighbourhoods of zero.

Soit un domaine d’intégrité intégralement clos dans son corps des quotients, et soit m un idéal premier de . Soit un domaine d’intégrité intégralement clos dans son corps des quotients et dont est sous-anneau; soit un idéal premier de tel que

Let Em denote the Euler number in the even suffix notation so that

(1.1)

where, as usual, after expansion of the left member Er is replaced by Er. Nielsen [4, p. 273] has proved that

(1.2)

Für jeden, der sich an die klassische Logik gewöhnt ist, mag undenkbar sein, dass der „Satz vom ausgeschlossenen Dritten” in der intuitionistischen Logik im allgemeinen nicht anerkannt ist. Aber es ist ganz natürlich, wenn man diejenige Tatsache beobachtet, dass eine äusserlich gleiche Aussage in der klassischen Logik und in der intuitionistischen doch ungleiche inhaltliche Sinne haben kann.

The boundary components of an abstract Riemann surface were defined by B. v. Kérékjértó [7] and utilized in the book [14] written by S. Stoïlow. It is the purpose of the present paper to investigate their images under conformal mapping and to solve the Dirichlet problem with boundary values distributed on them.

Recently Hing Tong [3], M. Katētov [4] and C. H. Dowker [2] have established two sorts of insertion theorems for semi-continuous functions defined on normal and countably paracompact normal spaces. The purpose of this paper is to give the insertion theorem for some typical semi-continuous functions defined on a T-space. The relations between Baire sets and Borel sets in some topological spaces are also studied.

In the present paper we shall remark that to each class of algebraically equivalent cycles on a Jacobian variety we can attach a symmetric element of the ring of endomorphisms of the variety and shall prove some formulae concerning attached symmetric elements.

Dans leurs mémoires classiques sur les groupes de transformations continus et infinis, MM. É. Cartan et E. Vessiot ont introduit la notion de prolongement des groupes de transformations pour définir l’isomorphisme et l’homomorphisme entre deux groupes de transformations.

It is a well known fact on Lorenz groups that a quadratic form f is definite if and only if the corresponding orthogonal group On(R∞, f) where R∞ is the real number field, is compact. In this note, we shall show that the analogue of this holds for the case of the p-adic orthogonal group On(Rp, f), where Rp is the rational p-adic number field, as a special result of the more general statement on the completely valued fields.

Cohomology theory for (associative) algebras was first established in general higher dimensionalities by G. Hochschild [3], [4], [5]. Algebras with vanishing 1-cohomology groups are separable semisimple algebras ([3], Theorem 4.1). On extending and refining our recent results [6], [8], [12], we establish in the present paper the following:

Let n ≧ 2. Let A be an (associative) algebra (of finite rank) possessing a unit element 1 over a field Ω, and N be its radical.

Usually the group structure in a homotopy group is defined directly and explicitly. But the algebraic approach to the topology, now common, seems to raise the following question : is that the only group sturcture which is natural from the algebraic topological point of view? On the other hand, several algebraists have begun to feel a necessity to construct a “homotopy or cohomotopy theory of groups,” and it may be allowed to say that one of the first steps to the problem is the axiomatization of homotopy groups. Our first question is of course a special case of the latter problem.

Let

(1)

be a word in two variables X, Y, i.e. an element in the free group F2 on two generators X, Y. Let us say that f defines an associative composition for a group G if for arbitrary elements a, b, c in G we have

(2)

where a ° b is defined by

(3)

The purpose of this note is to prove the following proposition, which is made use of in the preceding paper by Mr. M. Kuranishi in this number. First we shall explain the notations. Let be a word with respect to two letters, and let G be a free group generated by two free generators x and y.