Es sei f eine meromorphe nichtkonstante Funktion im Kreise |z|, < 1. Die Theorie des Verhaltens von f auf dem Rande des Kreises wird von zwei bekannten Sätzen beherrscht: der Satz von Riesz-Lusin-Priwaloff-Frostman-Nevanlinna, auf Grund dessen eine Menge aus |z| = 1, für die die Winkelgrenzwerte von f in einer Menge der Kapazität Null liegen, vom Lebesgueschen Masse Null ist und der Satz von Fatou-Nevanlinna, welcher besagt, dass, falls die Funktion f beschränktartig ist, dann hat sie fast überall auf |z| = 1 Winkelgrenzwerte. In vorliegender Arbeit werden diese zwei Sätze auf folgende Weise verallgemeinert. An Stelle des Kreises |z| < 1 betrachten wir eine beliebige Riemannsche Fläche R mit Greenscher Funktion, und an Stelle der meromorphen Funktion f nehmen wir eine nichtkonstante analytische Abbildung von R in einer Riemannschen Fläche R′(R′ offen oder kompakt, mit oder ohne Greenscher Funktion). Es sei Δ, bzw. Δ′, der ideale Rand von R, bzw. R′, im Sinne von Martin [11]; wir werden im Abschnitt IV eine Teilmenge der Menge Δ und eine Abbildung von (f) in R′ ∪ Δ′ definieren. Die Verallgemeinerung des Satzes von Fatou-Nevanlinna besteht in der Behauptung, dass, falls f eine Lindelöfsche Abbildung ist [6], so ist Δ-(f) von harmonischen Masse Null und die des Satzes von Riesz-Lusin-Priwaloff-Frostman-Nevanlinna darin, dass, wenn (A) eine polare Menge auf R′∪Δ′ ist, so ist die Menge A ⊂(f) vom harmonischen Masse Null.

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.

In 1958 D. Puppe developed the theory of Abbildungsfolge and applied it to the study of Hilton formula and his sphärenähnliche Mannigfaltigkeiten. We shall be concerned here with a dual situation.

When we consider modules A over a ring R which is not a commutative integral domain, the usual torsion theory becomes somewhat inadequate, since zero-divisors of R are disregarded and since the torsion elements of A do not in general form a submodule. In this paper we shall try to remedy such defects by modifying the fundamental notions such as torsion modules, divisible modules, etc.

In a Frobenius algebra A over a field K, there exists a linear function λ of A into K which does not map any proper ideal of A onto 0. Then the map φ : x → xφ where

defines an automorphism φ of A onto itself. This automorphism is called Nakayama’s automorphism. Now the following result is well known.

Our aim in this note is to generalize some topological results of commutative noetherian rings to non-commutative rings. As a supplemental remark of [2] we prove in § 1 that any right ideal of a complete right semi-local ring is closed, and that

for any finitely generated right module M over a complete right semi-local ring Λ where J is the Jacobson radical of Λ.

A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.

In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.

Notations. The following notations will be used throughout this paper.

˛: the identity of a finite group.

Q: the rational number field.

P: an algebraic number field of finite degree, fixed as the ground field.

l: a prime number.

ζl: a primitive l-th root of unity.

In the usual theory of harmonic functions on a plane domain, the fact that the boundary of the domain is realized relative to the complex plane plays an essential role and supplies many powerful tools, for instance, the solution of Dirichlet problem. But in the theory of harmonic functions on a general domain, i.e. on a Riemann surface, the main difficulty arises from the lack of the “visual” boundary of the surface. Needless to say, in general we cannot expect to get the “relative” boundary with respect to some other larger surface. In view of this, we need some “abstract” compactifications. It seems likely that we cannot expect to get the “universal” boundary which is appropriate for any harmonic functions since there exist many surfaces which do not admit some classes of harmonic functions as the classification theory shows. Hence we need many compactifications corresponding to what class of harmonic functions we are going to investigate.

Let D be a domain in the complex z-plane and γ be a boundary component of D consisting of a single point. The component γ is said to be weak if its image under any conformal mapping of D consists of a single point. If γ is not weak, then we say that γ is unstable (Sario [3], [4]).

Pontrjagin rings over the field of rational numbers of compact Lie groups are commutative in the sense of graded algebras (or anti-commutative in the classical terminology) [14]. Pontrjagin rings over the field Zp (p 0) of several compact simple Lie groups were studied by Borel [5]. The most examples are commutative. However, this is generally not true.