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.

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.

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).

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.

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.

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 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.

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

For each natural number n, let un(x)=(1—cos nx)/πnx2(xɛℝ). It is well–known that a bounded continuous function f on the real line ℝ is the Fourier transform of an integrable function on ℝ if and only if the functions Φn(f) (n= 1, 2,…), defined by

form a Cauchy sequence in the space L1(ℝ) (cf. [2]). Such a characterization, which can be extended to functions defined on a locally compact Abelian group more general than ℝ, is based on the fact that the space L1(ℝ) is complete with respect to convergence in mean.

The following general problem is of interest. Let Λ be an irreducible algebraic variety of degree d, in projective n-space Pn, defined over a field k; and suppose that K is a finite extension of k with [K: k] prime to d. If Λ has a point defined over K, then does it necessarily have a point defined over k?

It has been studied in various instances by several authors: see, for example, Cassels [2], Coray [3, 4], Pfister [5], Bremner, Lewis, Morton [1]. Coray [3] shows that a quartic curve Λ over Q may possess points in extension fields of Q of every odd degree greater than one, but have no points in Q itself. Some further examples of this instance occur in the paper of Bremner, Lewis, Morton, with the additional property that the curve Λ also possesses points in every p-adic completion Qp of Q.