Let R be an associative ring and f(x1,…, xd) a polynomial in noncommuting variables. We say that f is periodic or nil in R if for all r1,…, rd ∈ R we have that f(r1,…, rd) is periodic, respectively nilpotent (recall that a ∈ R is periodic if for some integer ).

Let be the least positive residue modulo 2tk of (2j- l)h. Define ut to be the number of with l≤j≤2t-2k such that . At the Special Session in Combinatorial Number Theory at the 1977 Summer AMS Meeting Szekeres [2] asked for a simple proof that if (h, 2k)=1, then

Here a simple proof will be given for the following equivalent result.

Let D be a division ring with center Z. Suppose for all xϵD, there exists a monic polynomial, fx(t), with integer coefficients such that fx(x)ϵZ. Then D is commutative.

Let F be a free group. Denote by the quotient group by the commutator subgroup which is a free abelian group. The fact that the natural map from Aut(F) into Aut() is an epimorphism, in case when F is finitely generated, was known as a consequence of the theory of Nielsen transformations ([2]) Proposition 4.4 and [3] Corollary 3.5.1).

With the aid of a recent result obtained by the first author, an expression is derived which unifies the well-known Jensen and Gould formulas.

The following question has been floating around for some time now and is also stated as Research Problem 26 in [4]:

Let R, S be unital rings and let 〈x〉 be an infinite cyclic group. Does R〈x〉≃S〈x〉 imply R≃S?

In this note, we present a collection of examples which answer the question in the negative. However, all of these examples consist of non-commutative rings, and the problem is still open in the case where R and S are assumed to be commutative.

Let (X, I, μ) be a σ-finite measure space and let T take Lp to Lp, p fixed, 1<p<∞,‖t‖p≤1. We shall say that the individual ergodic theorem holds for T if for any uniform sequence K1, k2,… (for the definition, see [2]) and for any f∊LP(X), the limit

exists and is finite almost everywhere.

In this paper, it is proved that a reduced ring R has an orthogonal completion if and only if for every idempotent e e R, eR has an orthogonal completion. Every orthogonal subset X of R has a supremum in Q max(R), the maximal two sided ring of quotients of R, and the orthogonal completion of a reduced ring R, if it exists, is isomorphic to a unique subring of Q max(R). Hence the orthogonal completion of a reduced ring R, if it exists, is unique upto isomorphism. A reduced ring R has an orthogonal completion if and only if the collection of those elements of Q max(R) which are supremums of orthogonal subsets of R form a subring of Q max(R). Furthermore, every projectable ring R has an orthogonal completion , which is a Baer ring. It is also proved that for projectable rings R, where is the idempotent filter of those dense right ideals of R which contain a maximal orthogonal subset of idempotents of R.

Van der Waerden [1, 4, 5] proved that if the nonnegative integers are partitioned into a finite number of sets, then at least one set in the partition contains arbitrarily long finite arithmetic progressions. This is equivalent to the result that a strictly increasing sequence of integers with bounded gaps contains arbitrarily long finite arithmetic progressions. Szemerèdi [3] proved the much deeper result that a sequence of integers of positive density contains arbitrarily long finite arithmetic progressions. The purpose of this note is a quantitative comparison of van der Waerden's theorem and sequences with bounded gaps.

Conditions for the existence and conjugacy of complements of certain minimal ideals of solvable Lie algebras over a Noetherian ring R are considered. Let L be a solvable Lie algebra and A be a minimal ideal of L. If L/A is nilpotent and L is not nilpotent then A has a complement in L, all such complements are conjugate and self-normalizing and if C is a complement then there exists an x∈L such that C = {y∈L; yadnx = 0 for some n = 1, 2,…}. A similar result holds if A is self-centralizing and a finitely generated R-module.

We give a simple proof of a theorem by Andrunakievič and Rjabuhin which states that a reduced ring is a subdirect product of entire rings. Our proof makes no use of m-systems and is in some sense similar to the proof of the corresponding theorem in the commutative case due to Krull.

A reduced ring is a ring without non-zero nilpotent elements. It is wellknown that if a reduced ring is commutative, then it is a subdirect product of integral domains [2]. This result has been generalized to arbitrary reduced rings [1]. The proof in the general case is somewhat complicated. We present a simple proposition which leads to a simple proof of the general case.

If M is a simply-connected 4-manifold with boundary, let D(M) denote its double MU∂M(-M). If M is closed, let D(M) denote M#-M. In either case, D(M) is a simply-connected 4-manifold of index zero, and so by a theorem of Wall [8], M#k(S2xS2) must be standard for k sufficiently large, where by standard we mean diffeomorphic to the connected sum of copies of S2 x S2 and S2×S2, the non-trivial S2 bundle over S2 (which is itself diffeomorphic to ℂP2#-ℂP2 [7]). In this paper we give abound on k, in the case where M has no 3-handles.

An example is given of a topological space that does not admit a strongly complete quasi-uniform structure.

In a paper with the above title, T. M. Apostol and S. Chowla [1] proved the following result:

Theorem 1.For relatively prime integers h and k, l ≤ h ≤ k, the series

does not admit of an Euler product decomposition, that is, an identity of the form

1

except when h = k = l; fc = 1, fc = 2. The series on the right is extended over all primes p and is assumed to be absolutely convergent forR(s)>1.

In his recent book [3] Nadler observes that the property of admitting a Whitney map is of fundamental importance in studying the internal structure of hyperspaces, especially their arc structure. Nadler presents three distinct methods of constructing a Whitney map on the hyperspace 2X of nonempty closed subsets of a continuum.

A partially ordered space is a topological space X endowed with a partial order ≤ whose graph is a closed subset of X×X. It is well-known (see, for example, [2], page 167) that if X is a regular Hausdorff space then 2X is a partially ordered space with respect to inclusion.

Let Hp(0<p≤∞) be the Hardy class of functions holomorphic in D={|z|<1}, and let f be a function holomorphic in D. It follows from the well known theorem [1, Theorem 3.11, p. 42] that if (I) , then (II) f can be extended continuously to the closed disk {|z|≤1}, and consequently, (III) f is normal in D in the sense of O. Lehto and K. I. Virtanen [2]. Since there is a considerable gap between (II) and (III), one would suspect that the condition (I) might be much stronger than necessary to obtain the conclusion (III). However, it is curious that the implication (I)⇒(III) is sharp as the following theorem shows.

If ϕ(n) denotes the Euler function, for n = p a prime we have ϕ(n)/n = (1-1/p), which implies that

In this note we consider a refinement of this result. Namely, we prove that

1

where P∗(k) is the largest integer of the form where p1 < p2<…<pr are the first r primes in ascending order.

Let be a linear partial differential operator with C∞- coefficients. The study of P(∂) as an operator in L2(ℝn) usually starts with the investigation of the minimal operator P0 which is the closure of P(∂) acting on . In the case of constant coefficients it is known that the domain D(P0) of P0 at least contains the space (cf. Schechter [4, p. 58, Lemma 1.2]).