In the study of idealizer rings in [14], the global dimensions of certain subrings were calculated. This work was extended by Goodearl [9]. Here it is shown, in Section 1, that the techniques involved can be used rather more generally than was apparent there. The results concern a ring S with a right ideal A such that SA = S, and a subring R of S containing A. They describe gl dim R in terms of gl dim S and the projective dimensions of simple subfactors of S/R or R/A, provided that these R-modules are sufficiently well conditioned.

We give a complete description of the Brown–McCoy radical of a semigroup ring R[S], where R is an arbitrary associative ring and S is a commutative cancellative semigroup; in particular we obtain the answer to a question of E. Puczyłowski stated in [11]

Throughout this note all rings R are associative with unity 1; all semigroups S are commutative and cancellative with unity. Note that the condition that R and S have a unity can be dropped (cf. [8]). The quotient group of S is denoted by Q(S). We say that S is torsion free (resp. has torsion free rank n) if Q(S) is torsion free (resp. has torsion free rank n). The Brown–McCoy radical (i.e. the upper radical determined by the class of all simple rings with unity) of a ring R is denoted by u(R). We refer to [2] for further detail on radicals and in particular on the Brown–McCoy radical.

First we state some well-known results and a preliminary lemma. Let R and T be rings with the same unity such that R ⊂ T. Then T is said to be a normalizing extension of R if T = Rx1+…+Rxn for certain elements x1, …, xn of T and Rxi = xiR for all i such that 1 ∨i∨n. If all xi are central in T, then we say that T is a central normalizing extension of R.

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f then

where ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.

Let denote the unit ball in ℂ2 and let Sdenote its boundary, the unit sphere. For z ∈ B and δ>0, the following non isotropic balls are defined, where

A finite positive Borel measure μ, on B is called a Carleson measure if there exists a constant C for which

Here σ denotes normalized surface area measure on S. The following theorem was obtained by Hörmander [6] as a special case of more general variants for strictly pseudoconvex domains in ℂn. Recently Cima and Wogen [3] derived it from a Carleson measure theorem for Bergman spaces of the ball. A different direct approach to the Bergman context, and related settings, is given in Leucking [7].

An algebra (L; ν, ^, *, +, 0, 1) of type (2, 2, 1, 1, 0, 0) is a distributive double p-algebra provided (L; ν, ^, 0, 1) is a distributive (0, l)-lattice, and *, + are unary operations of pseudocomplementation, or dual pseudocomplementation, respectively: the operation * satisfies x<a* if and only if x^a = 0, while x>a+ holds if and only if xνa = 1.

The concept of superdiagonal forms for n × nmatrices T with complex entries has been extended by J. R. Ringrose [4] to the setting of compact linear operators T:X→X acting on a complex Banach space X. In a recent paper D. Koros [2] generalized Ringrose's approach to the case of compact linear operators T:X→X on a complex locally convex space X. The reason why both authors confine their attention to the class of compact linear operators is that the existence of proper closed invariant subspaces is, aside from Riesz-Schauder theory, the main tool in their construction. In the present paper it is shown that the existence of superdiagonal forms possesses a certain permanence property in the following sense.

C. Eberhart and W. Williams [3] showed that the least inverse semigroup congruence , on an orthodox semigroup S, plays a very important role in determining the structure of the lattice of congruences on S. In this note we show that their results can be applied to give an explicit construction for the idempotent separating congruences on S in terms of idempotent separating congruences on S/.

Let C= CmXCmX…XCm be the finite abelian group of order mn generated by n elements w1…,wn of order m. Let C be the field of complex numbers and P a projective representation of G with factor set α over C (see Morris [2]). Further let

and

Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let K(H) denote the ideal of compact operators on H. For any compact operator A let |A|=(A*A)1,2 and S1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeatedaccording to multiplicity. If, for some 1<p<∞, si(A)p <∞, we say that A is in the Schatten p-class Cp and ∥A∥p=1/p is the p-norm of A. Hence, C1 is the trace class, C2 is the Hilbert–Schmidt class, and C∞ is the ideal of compact operators K(H).

Let X be a bordered Klein surface, by which we mean a Klein surface with non-empty boundary. X is characterized topologically by its orientability, the number k of its boundary components and the genus p of the closed surface obtained by filling in all the holes. The algebraic genus g of X is defined by

.

If g≥2 it is known that if G is a group of automorphisms of X then |G|≤12(g-l) and that the upper bound is attained for infinitely many values of g ([4], [5]). A bordered Klein surface for which this upper bound is attained is said to have maximal symmetry. A group of 12(g-l) automorphisms of a bordered Klein surface of algebraic genus g is called an M*-group and it is known that a finite group G is an M*-group if and only if it is generated by 3 non-trivial elements T1, T2, T3 which obey the relations

([4]).

The problem of determining whether or not two operators are unitarily equivalent has been around for many years and considerable work has been done in attempting to solve this problem (see for example [1], [3], [5], [6], [7], [10], [11], [12], [15], [16], [17], [18], [19], [20], [21] and [22]). In many cases, a complete set of unitary invariants is furnished for a certain class of operators. Here we just mention two of such results which are related to what we are going to discuss. The first one was due to Arveson, who showed that two irreducible compact operators are unitarily equivalent if and only if they have the same nth algebraic matricial ranges, for each n≧1 ([1] and Theorem 2.4.3 of [3]). The second one was due to Parrott, who showed that two compact operators with zero reducing null spaces are unitarily equivalent if and only if they have the same nth spatial matricial ranges, for each n≧1 ([5, p. 146]). In this paper, we investigate the closures of the spatial matricial ranges of compact operators and obtain a complete set of unitary invariants for compact operators, from which Parrott's result follows easily.

In 1959, Bishop [4] published a seminal paper in which he studied various types of spectral decompositions or “duality theories” that an arbitrary bounded linear operator on a reflexive Banach space might have. In the course of his investigations, he isolated the following analytic property which he called condition (β).

Let H be a complex Hilbert space. For any operator (bounded linear transformation) T on H, we denote the spectrum of T by σ(T). Let T = (T1, …, Tn) be an n-tuple of commuting operators on H. Let Sp(T) be the Taylor joint spectrum of T. We refer the reader to [8] for the definition of Sp(T). A point v = (v1, …, vn) of ℂn is in the joint approximate point spectrum σπ(T) of T if there exists a sequence {xk} of unit vectors in H such that

.

A point v = (v1, …, vn) of ℂn is in the joint approximate compression spectrum σs(T) of T if there exists a sequence {xk} of unit vectors in H such that

A point v=(v1, …, vn) of ℂn is in the joint point spectrum σp(T) of T if there exists a non-zero vector x in H such that (Ti-vi)x = 0 for all i, 1 ≤ j ≤ n.

If the group G=AB is the product of two abelian subgroups A and B, then G is metabelian by a well-known result of Itô [8], so that the commutator subgroup G' of G is abelian. In the following we are concerned with the following condition:

There exists a normal subgroup

which is contained in A or B.

Recently, Holt and Howlett in [7] have given an example of a countably infinite p-group G = AB, which is the product of two elementary abelian subgroups A and B with Core(A) = Core (B) = 1, so that in this group (*) does not hold. Also, Sysak in [13] gives an example of a product G = AB of two free abelian subgroups A and B with Core(A)=Core(B)=l.

K. Shoji has pointed out to me that construction [1] does not always yield a completion. In the notation of [1], the homomorphism from the strong semilattice of cancellative semigroups S to its purported completion T in Abian's order is not always a monomorphism. The difficulty arises when there is eɛ E, e=sup{e'ɛEe'<e>e} but {‐e,e'}e' is not faithful, i.e. there are x, y with x¬y in Se such that φe,e'(x)=φe,e'(y) for all e'<e. A modification of the construction saves all parts of Theorem 1 except the fact that the new embedding S⊆T need not preserve suprema existing in S; it does if S is a semilattice of groups. The sequel [2] also needs amodification in the form of an additional hypothesis.

Let A be a commutative Noetherian ring (with non-zero identity). The Cousin complex C(A) for A is described in [6, §2]: it is a complex of A-modules and A-homomorphisms

with the property that, for each n≥0,

Cohen-Macaulay rings may be characterized in terms of the Cousin complex: A is a Cohen-Macaulay ring if and only if C(A) is exact [6, (4.7)]. Also the Cousin complex provides a natural minimal injective resolution for a Gorenstein ring: see [6, (5.4)].

In this paper we study the asymptotic behaviour of the following systems of ordinary differential equations:

where the identically zero function is a solution of (N) i.e. f(t, 0)=0 for all time t. Suppose one knows that all the solutions of (N) which start near zero remain near zero for all future time or even that they approach zero as time increases. For the perturbed systems (P) and (P1) the above property concerning the solutions near zero may or may not remain true. A more precise formulation of this problem is as follows: if zero is stable or asymptotically stable for (N), and if the functions g(t, x) and h(t, x) are small in some sense, give conditions on f(t, x) so that zero is (eventually) stable or asymptotically stable for (P) and (P1).

In recent years versions of the Lebesgue and the Hewitt-Yosida decomposition theorems have been proved for group-valued measures. For example, Traynor [4], [6] has established Lebesgue decomposition theorems for exhaustive groupvalued measures on a ring using (1) algebraic and (2) topological notions of continuity and singularity, and generalizations of the Hewitt-Yosida theorem have been given by Drewnowski [2], Traynor [5] and Khurana [3]. In this paper we consider group-valued submeasures and in particular we have established a decomposition theorem from which analogues of the Lebesgue and Hewitt-Yosida decomposition theorems for submeasures may be derived. Our methods are based on those used by Drewnowski in [2] and the main theorem established generalizes Theorem 4.1 of [2].

A problem of special interest in the study of automorphism groups of surfaces are the bounds of the orders of the groups as a function of the genus of the surface.

May has proved that a Klein surface with boundary of algebraic genus p has at most 12(p–1) automorphisms [9].

In this paper we study the highest possible prime order for a group of automorphisms of a Klein surface. This problem was solved for Riemann surfaces by Moore in [10]. We shall use his results for studying the Klein surfaces that are not Riemann surfaces. The more general result that we obtain is the following: if X is a Klein surface of algebraic genus p, and G is a group of automorphisms of X, of prime order n, then n ≤ p + 1.

In this note we consider the question as to which central division algebras occur as the Clifford algebra of a quadratic form over a field. Non-commutative ones other than quaternion division algebras can occur and it is also the case that there are certain central division algebras D which, while not themselves occurring as a Clifford algebra, are such that some matrix ring over D does occur as a Clifford algebra. We also consider the further question as to which involutions on the division algebra can occur as one of two natural involutions on the Clifford algebra.