In this paper, we continue the study of integral extensions begun in [7]. Whereas in the previous paper, we dealt exclusively with the extension A[x]/(α(x)), α(x) a monic polynomial over A, we now deal with arbitrary integral extensions. Applications of the results presented herein will be made in subsequent papers.

To simplify our presentation, we make the following conventions. By an algebra, we will always mean a commutative complex algebra with an identity element, usually denoted by e

In a finite graph with no loops nor multiple edges, two points a and b are said to be connected by an r-way, or more explicitly, by a line r-way a — b if there are r paths, no two of which have lines in common (although they may share common points), which join a to b. In this note we demonstrate that any graph with n points and 3n — 2 or more lines must contain a pair of points joined by a 6-way, and that 3n — 2 is the minimum number of lines which guarantees the presence of a 6-way in a graph of n points.

In the language of [3], this minimum number of lines needed to guarantee a 6-way is denoted U(n). For the background of this problem, the reader is referred to [3].

In 1958, G. D. Findlay and J. Lambek defined a relationship between three R-modules, A ≦ B(C), to mean that A ⫅ B and every R-homomorphism from A into C can be uniquely extended to an irreducible partial homomorphism from B into C. If A ≦ B(B), then B is called a rational extension of A and in [5] it is shown that every module has a maximal rational extension in its injective hull which is unique up to isomorphism. A module is called rationally complete provided it has no proper rational extension.

Consider a positive integer r > 1. We say that the rth root problem is solvable for a group G if we can decide for any W ॉ G whether or not W has an rth root, i.e. whether or not there exists V ॉ G such that W = Vr.

Baumslag, Boone and Neumann [1] proved that there exists a finitely presented group with all root problems unsolvable. Here we are concerned with the relationship between the different root problems.

In this paper, we extend the class of spaces to which the Σ and β theorems of Hodel apply, as well as the sum and subset theorems of [2]. Instead of the open cover definition of countable paracompactness, we utilize an equivalent formulation of countable paracompactness, due to Ishikawa [3].

Let R be an associative ring with identity, and let denote the category of unital left R-modules. It is known that if R is a commutative, Noetherian ring, then the maximal torsion radicals of correspond to the minimal prime ideals of R. In fact, Nӑstӑsescu and Popescu [15, Proposition 2.7] have given a more general result valid for arbitrary commutative rings. This paper investigates maximal torsion radicals over rings not necessarily commutative.

Let (respectively ) denote the ring of germs at 0 ∈ Rn of all C∞ functions (respectively Cμ functions) from Rn to R. For a given where is the space of all germs of C∞ mappings Rn → Rp, let J(φ) denote the ideal in generated by φ1, … , φp and the Jacobian determinants

where Let

Clearly, is an ideal in and where is the (unique) maximal ideal of .

Si S est un espace topologique compact séparé et si ϕ : S → S est une fonction continue, l'opérateur: A : C(S) → C(S), défini par Ag = g o ϕ est linéaire positif tel que Ae = e (sur C(S) nous considérons le cône usuel et e désigne la fonction identiquement 1 sur S).

Let K[G] denote the group ring of G over the field K. One of the interesting problems which arises in the study of such rings is to find precisely when they satisfy polynomial identities. This has been solved for char K = 0 in [1] and for char K = p > 0 in [3]. The answer is as follows. If p > 0 we say that group A is p-abelian if A', the commutator subgroup of A, is a finite p-group. Moreover, for convenience, we say A is 0-abelian if and only if it is abelian.

Familiarity with the basic notions of graph theory is assumed. Loops and multiple edges are not permitted. An orientation of an edge e of a graph G is a designation of one of the ends of e as the positive end and the other as the negative end. We say that e is oriented from the positive end to the negative end. If e joins vertex v to vertex w and v is the positive end of e, we write e = (v, w). An orientation of G is a set of orientations, one for each edge of G; a graph with an orientation is called a directed graph.

Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).

It has been proved [3, p. 630] that the martingale convergence theorem obtained by Andersen and Jessen [1, p. 5] follows from the classical theory developed by Doob. By using some results of Yosida and Hewitt [9] on finitely additive set functions, Johansen and Karush [7] proved that the identification of the limit function as a derivative in the approach of Andersen and Jessen can be obtained in the general case. In this paper we sharpen the methods of Andersen and Jessen to obtain a ratio limit theorem for “approximate martingales”.

In this paper we are chiefly concerned with the chromatic sums we have called l and h, with colour-number 3. In this case h can be interpreted as enumerating the rooted Eulerian triangulations with a given number of faces, and l as enumerating such triangulations with a given number of faces and a given valency for the root-vertex. The series h has been determined already, by summation from the formula enumerating even slicings [3]. However our formula for l does not seem to have been published before, though it could presumably be derived in a similar way.

A space X is locally planar if each point of X has a neighborhood which is embeddable in the plane. If X is a closed, locally planar subset of E3, we will say that X is locally tame if each point of X has a neighborhood in X which lies on a tame disk in E3; if every cell-like subset of X has such a neighborhood, we say that X is strongly locally tame.

An invertible measure preserving transformation T on the unit interval I generates a unitary operator U on the space L2(I) of Lebesque square integrable functions given by (Uf)(x) = f(Tx) for all f in L2(I) and x in I. By definition

for all f , g in L2(I), the bar denoting complex conjugation.

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.

If A is a finite matrix with complex elements, and if A = AT (where AT denotes the transpose of A ), it is known (see [8] ) that there exists a unitary matrix U such that UA UT = D is a real diagonal matrix with non-negative elements which is a canonical form for A relative to the given U, UT transformation.

Although the notion of the maximal quotient ring of a nonsingular ring has been around for some time, not much is known about its structure in general beyond the important theorems of Johnson and Utumi [4; 11] that it is von Neumann regular and self-injective. The purpose of this paper is to study the structure of such a regular, self-injective ring R by looking at its prime ideals. Initially, we show that the primes of R separate into two types, called ‘'essential” and ‘“closed”, and that for any prime P, the two-sided ideals in the ring R/P are linearly ordered.

An unsolved problem in metrization theory is whether every Hausdorff M-space with a Gg-diagonal is metrizable. There are several recent results which have a bearing upon this question. In [9], P. Zenor showed that an M-space is metrizable if and only if it has a regular Gδ-diagonal; in [1], Borges showed that a regular meta-Lindelöf M-space is metrizable if and only if it has a -diagonal.

The purpose of this paper is two-fold : first, to show that Dedekind domains satisfy a generalization of the Wedderburn-Mal'cev Theorem and, secondly, to classify certain types of finitely generated projective algebras over a Dedekind domain.

With respect to the first problem, E. C. Ingraham has shown that a Dedekind domain R is an inertial coefficient ring (IC-ring) if and only if R has zero radical or R is a local Hensel ring.