Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an M-primary ideal of a local ring (R, M) satisfying the condition that Q: M = Q + Ms−1 where s is the index of Q, then all irreducible components of Q have index s. (Q is “index-unmixed”.) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples.

On a set X, let μ* be an outer measure and μ the measure induced by μ*. We show that if X is a finite set, then the measure μ is saturated. We give two examples of non-regular outer measures on an infinite set X which induce non-saturated and saturated measures, respectively. These answer a query posed by Wilansky.

This paper demonstrates a connection between two measures of discrepancy of sequences which arise in the theory of uniform distribution modulo one. The sequence formed from the non-negative integer multiples of an irrational number ξ is investigated and, by an application of the “Steinhaus Conjecture”, some values of the two discrepancies are obtained using continued fractions.

A result of Howson is that two finitely generated subgroups U and V of a free group have finitely generated intersection. Hanna Neumann showed further that, if m, n and N are the ranks of U, V and U ∩ V respectively, then N ≤ 2(m−1)(n−1) + 1, and Burns strengthened this, showing that N ≤ 2(m−1)(n−1) − m + 2 (if m ≤ n). This paper presents a new and simple proof of Burns' result. Further, the graph-theoretical ideas used provide still stronger bounds in certain special cases.

This paper provides examples of indecomposable 2-groups with non-abelian automorphism groups in which every automorphism is central.

Let KG be the group ring of the group G over the infinite field K, and let U(KG) be its group of units. If G is torsion, we obtain necessary and sufficient conditions for a finite subgroup H of G to be either normal or subnormal in U(KG). Actually, if H is subnormal in U(KG), we can handle not only the case H finite, but the precise assumptions depend on the characteristic of K.

In this note a theorem proved by Abu-Khuzam and Yaqub has been improved as follows: let R be a semi-simple ring such that for all x, y in R there exists a positive integer n = n(x, y) for which either (xy)n + (yx)n or (xy)n – (yx)n is central. Then R is commutative.

We introduce a new class of generalized arcwise connected functions and discuss their basic properties. Our generalization is illustrated by an example and an application is given for a mathematical programming problem involving this new class of functions.

Let L denote the subvariety lattice of the variety of distributive double p-algebras, that is, the lattice whose universe consists of all varieties of distributive double p-algebras and whose ordering is the inclusion relation. We prove in this paper that each proper filter in L is uncountable. Moreover, we prove that except for the trivial variety (the zero in L) and the variety of Boolean algebras (the unique atom in L) every other element of L, generated by a finite algebra, has infinitely many covers in L, among which at least one is not generated by any finite algebra. The former result strengthens a result of Urquhart who showed that the lattice L is uncountable. On the other hand, both of our results indicate a high complexity of the lattice L at least in comparison with the subvariety lattice of the variety of distributive p-algebras, since a result of Lee shows that the latter lattice forms a chain of type ω + 1 and every cover in it of the variety generated by a finite algebra is itself generated by a finite algebra.

The purpose of this paper is to study the trajectory multifunction Φ(·) determined by the indefinite set valued integral of a measurable Banach space valued multifunction F(·), that is for all t ∈ [0, T], , where the set valued integral is interpreted in the sense of Aumann. We study the topological and algebraic properties of SΦ equaling the set of selectors of Φ(·) whose primitive is an integrable selector of F(·). We also determine several useful properties that Φ(·) possesses and finally we present some convergence and stability results using the Kuratowski-Mosco convergence of sets.

It is proved here that for two distribution functions with equal moments up to order n, the number of crossings is at least n, and if exactly n, the remaining odd or even moments (for n even or odd respectively) do not cross again. This both generalises and extends a number of previous results.

Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

This note completes the determination of the basis numbers of the complete bipartite graphs.

It is shown that expanding self-maps of the interval with a finite number of turning points must have periodic points whose periods are not integral power of 2 and therefore are chaotic.

Shimizu, Aiyoshi and Katayama have recently given a finite dimensional generalization of the classical Farkas Lemma. In this note we show that a result of Pshenichnyi on convex programming can be used to give a generalization of the result of Shimizu, Aiyoshi and Katayama to infinite dimensional spaces. A generalized Farkas Lemma of Glover is also obtained.

In this paper some results of Egorov's theorem type are given for functions with values in locally convex spaces and Riesz's theorem is proved for functions taking values in a sequentially complete locally convex space.