Let be a family of subsets of a topological space X. We do not require to be a covering of X, nor do we assume that the members of are necessarily open. In this paper we shall assume that is of a special sort, which we call Σ-Finite. We show that a Σ-Finite family is both locally finite and star-finite, and in particular that an open covering of X is Σ-Finite if and only if it is star-finite. We then prove that every Σ-Finite family is ᓂ-discrete, so that in particular, every star-finite open covering of X is (ᓂ-discrete. There seems to be some applications of this fact.

All rings are associative with unity. A ring R is prime if xRy ≠ 0 whenever x and y are nonzero. A ring R is (left) primitive if there exists a faithful irreducible left R-module.

If the group ring R[G] is primitive, what can we say about R? First, since every primitive ring is prime, we know that R is prime, by the following

THEOREM 1 (Connell [1, 675]). The group ring R[G] is prime if and only if R is prime and G has no non-trivial finite normal subgroup.

Let MR be a right module over the associative ring R (with 1). Assume one has an expression for M as a colimit (direct limit) of a system

over the (directed) poset D. A natural way to get M as a colimit of the family ﹛F∞ —” Fβ|∞, fβ ∈ E﹜ for some subset £ of D is to take E cofinal in D. However, if one is concerned about the cardinality of the set E, cofinal subsets may be too large. Let us look at a specific example. Lazard [3] has shown that any flat MR is a direct limit of finitely generated free R-modules. The cardinality of his indexing set depends on the cardinality of M.

On étudie certains sous-espaces de l'algèbre de Banach Bp(G) des multiplicateurs de AP(G). On montre entre autres que l'espace des coefficients d'une représentation de G sur un espace Lp n'est pas nécessairement fermé dans Bp(G) à moins que p soit égal à deux.

Consider the wth order scalar ordinary differential equation

with pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:

DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decomposition

where M1 and M2 are subspaces of X such that

(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;

(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

Let R be a commutative ring. A bilinear space (E, B) over R is â finitely generated projective R-module E together with a symmetric bilinear mapping B:E X E →R which is nondegenerate (i.e. the natural mapping E → HomR(E﹜ R) induced by B is an isomorphism). A quadratic space (E, B, ) is a bilinear space (E, B) together with a quadratic mapping ϕ:E →R such that B(x, y) = ϕ (x + y) — ϕ (x) — ϕ (y) and ϕ (rx) = r2ϕ (x) for all x, y in E and r in R. If 2 is a unit in R, then ϕ (x) = ½. B﹛x,x) and the two types of spaces are in obvious 1 — 1 correspondence.

In this paper we consider matrix representations of compact groups over the field of the complex numbers. We shall deal mainly with finite groups.

The Kronecker product of two irreducible representations σ1 and σ2 of a group is in general a reducible representation of . The explicit reduction of such a product to irreducible representations σ3 can be performed by means of a unitary matrix, the elements of which are called Wigner coefficients or Clebsch-Gordan coefficients [1; 25; 27].

Introduction. All rings occurring are associative and possess a unity, which is preserved under subrings and ring homomorphisms. All modules are unitary right modules. We denote the category of rights-modules.

In recent years several authors have studied rings R by imposing restrictions on the torsion theories [4] of . (See for instance [2; 23; 24].) This paper offers another alternative to that trend, namely the study of rings R via their set of kernel functors K﹛R).

1. Introduction. We prove, using a theorem of M. Hall on cyclic projective planes, that if g is a prime power such that either 1 + q + q2 is a prime congruent to 3, 5 or 7 (mod 8) or 3 + 2q + 2q2 is a prime power, then there exists a skew-Hadamard matrix of the Goethals-Seidel type of order 4(1 + q + q2). (A Hadamard matrix H is said to be of skew type if one of H + I, H — lis skew symmetric. ) If 1 + q + q2 is a prime congruent to 1 (mod 8), then a Hadamard matrix, not necessarily of skew type, of order 4(1 + q + q2) is constructed. The smallest new Hadamard matrix obtained has order 292.

Let K be a field and Mn﹛K) denote the vector space of n X n matrices over K. Marcus [4] posed the following general problem: Let W be a subspace of Mn(K) and S a subset of W. Describe the set L(S, W) of all linear transformations T on W such that T(S) is contained in S.

In [14, §4], some theorems were obtained about generalized polynomial identities in rings with involution, but the statements had to be weakened somewhat because a structure theory of rings with involution had not yet been developed sufficiently to permit proofs to utilize enough properties of rings with involution. In this paper, such a theory is developed. The key concept is that of the central closure of a ring with involution, given in § 1, shown to have properties analogous to the central closure of a ring without involution. In § 2, the theory of primitive rings with involution, first set forth by Baxter-Martindale [5], is pushed forward, to enable a setting of generalized identities in rings with involution which can parallel the non-involutory situation.

The graphs considered here are finite and have no loops or multiple edges. In particular, Km denotes the complete graph on m vertices. For any graph G,V(G) and E(G) denote, respectively, the vertex and edge sets of G. A forest is a graph which has no cycles and a tree is a connected forest. The reader is referred to [1] or [4] for the meaning of terms not defined in this paper.

Let the standard identity of degree m be given by

Then we shall show that the set of all n X n skew-symmetric matrices over a field of characteristic 0 satisfies the standard identity of degree at least 2n — 2; specifically, we shall prove the following.

Let M be a. complete connected Riemannian manifold of dimension n and let £:M → Rn+k be an isometric immersion into the Euclidean space Rn+k. Let ∇ be the connection on Mn and let be the Euclidean connection on Rn+k. Also let

denote the second fundamental form B(X, Y) = (xY)→. Here TP(M) denotes the tangent space at p, NP(M) the normal space and (…)→ the normal component.

Let R be an associative ring with unit element. Mod-R and R-Mod will denote the categories of unitary right and left R-modules, respectively, and all modules are assumed to be in Mod-R unless otherwise specified. For all M, N ϵ Mod-R, HomR(M, N) will usually be abbreviated as [M, N]. For the definitions of basic terms, and an exposition on torsion theories in Mod-R, the reader is referred to Lambek [6]. Jans [5] has called a class of modules which is closed under submodules, direct products, homomorphic images, group extensions, and isomorphic images a TTF (torsion-torsionfree) class.

Let R be a simple ring, of characteristic not 2, having an involution *. Let 5 = ﹛x G R|x* = x﹜ and K = ﹛x £ R|x* = — x﹜ be the set of symmetric and skew elements, respectively, of R.

In [1] we discuss the structure of S as a Jordan ring and K as a Lie ring. In [2] we considered cross-over submodules, namely additive subgroups U ⊂ K, V ⊂ S such that

A finite partially ordered set (poset) P is customarily represented by drawing a small circle for each point, with a lower than b whenever a < b in P, and drawing a straight line segment from a to b whenever a is covered by b in P (see, for example, G. Birkhoff [2, p. 4]). A poset P is planar if such a diagram can be drawn for P in which none of the straight line segments intersect.

W. A. J. Luxemburg and A. C. Zaanen in [7] and W. A. J. Luxemburg in [5] have studied the order properties of the order bounded linear functionals of a given Riesz space L. In this paper we consider the vector space (L, M) of the order bounded linear transformations from a given Riesz space L into a Dedekind complete Riesz space M.

For any radical property Q, a nonzero simple ring (all rings in this paper are assumed to be associative) must make up its mind so to speak and must be either Q radical or Q semi-simple. Every Q thus divides the class of all nonzero simple rings into two disjoint classes. Conversely any partition of the nonzero simple rings into two disjoint classes leads to at least two radicals [1, p. 16].