An inverse semigroup whose idempotents form an ω-chain e0 > e1 > e2 > … is called briefly an ω-semigroup. A structure theorem for simple ω-semigroups was established by Kočin [7]; a related structure theorem for simple, and also general, ω-semigroups was proved by Munn [10]. These results represent an extension of the structure theorem for bisimple a ω-semigroups due to Reilly [14].

The use of graphs in the study of groups is well-established. In this paper, we wish to indicate how certain graph-like objects may be used in a similar way. A diagram is a pseudograph which may have some free edges, i.e. edges with just one end.

Let G be a finite additive abelian group, and suppose that A and B are subsets of G. We say that G = A⊕B if every element g ∈ G can be uniquely written in the form g = a + b, where a ∈ A, b ∈ B. The study of such decompositions (usually called factorizations in the literature) was initiated by G. Hájos [3] in connection with his solution to a problem of Minkowski in the geometry of numbers.

The purpose of this paper is to show the equivalence of convergence, an associated projective system of measures and a martingale decomposition for a uniformly integrable stochastic process. Emphasis is placed on a direct juxtaposition of these concepts and on displaying underlying mechanisms.

The impact of the martingale convergence theorem on contemporary probability theory has been immense. Therein lies the reason for numerous generalizations of both the basic martingale convergence theorem and the martingale concept itself.

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

Questions about polynomials can be turned into questions about matrices by associating with the polynomial

(over an arbitrary field) its companion matrix

which has p/an as its characteristic polynomial. This technique is often used in stability theory, as indicated in [1]; companion matrices also occur in the theory of the rational canonical form.

In this paper we continue the investigation of minimal conditions on semigroups begun by J. A. Green [3] and taken up by Munn [5]. A unified account of the results in [3] and [5], together with some additional material, is presented in the text-book by Clifford and Preston [1, §6.6]. All terminology and notation not introduced explicitly willbe as in [1].

The following two results in the theory of division algebras are well known and easily proved, for an arbitrary commutative field k (cf. for example [3, Chapter 10]).

(i) The tensor product of two central division algebras over k of coprime degrees is again a division algebra.

(ii) Every central division algebra over k is a tensor product of division algebras of prime power degrees.

It is natural to ask whether corresponding results hold for commutative fields. The answers are not hard to find but (as far as I am aware) have not appeared in print before; since they throw some light on the nature of tensor products they seemed worth recording.

If f is an entire function in the complex plane such that

where 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).

Let f denote a function, meromorphic in C. The question of when a deficient value of f, in the sense of Nevanlinna, is an asymptotic value has recently received some attention (see e.g. Hayman [6]). We assume acquaintance with the standard notation of the Nevanlinna theory ([[5] Chapter I) which we use without further mention. The following two theorems are known ([1] Theorem 4, and [6] Corollary 2).

A (bounded, linear) operator H on a Banach space is said to be hermitian if ∥exp(itH)∥ = 1 for all real t. An operator N on is said to be normal if N = H + iK, where H and K are commuting hermitian operators. These definitions generalize those familiar concepts of operators on Hilbert spaces. Also, the normal derivations defined in [1] are normal operators. For more details about hermitian operators and normal operators on general Banach spaces, see [4]. The main result concerning normal operators in the present paper is the following theorem.

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.