The intent of this paper is to study a class of algebras which do not necessarily obey the association law but instead obey a law which bears a marked resemblance to associativity. For lack of a better name we call this class the class of scalar dependent algebras. Specifically, an algebra A over a field F is called scalar dependent if there is a map g: A × A × A → F such that (xy)z = g(x, y, z)x(yz), for all x, y, z in A. To obtain our results we shall assume throughout that A is a scalar dependent algebra with an identity element e over a field of characteristic not 2 satisfying

(I) (x, x, x) = 0.

As usual, the associator (x,y,z) is defined by (x,y,z) = (xy)z — x(yz). An example is given to show that (I) is not implied by scalar dependency.

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.

In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

One of the most useful properties of a compact Hausdorff space is that such a space is closed whenever embedded into a Hausdorff space. This property does not extend to compact spaces with respect to embeddings into arbitrary spaces. Thus, an interesting topological problem is to characterize the types of absolute “closure” properties that are possessed by compact spaces. This is the problem that is solved in the present paper.

The following notation and terminology will be used below. We shall consider a fixed space X and subspace A, representing arbitrary nonempty open subsets of X (respectively A ) by W (respectively V).

A group G is right-ordered if it can be totally ordered so that for any a, b, c in G, a < b implies that ac < bc. Right-ordered groups, considered as order preserving automorphisms of ordered sets, were studied by Cohn in [4]; but the first systematic study of the structure of these groups was made by Conrad in [5] where he gave several natural characterizations of right-ordered groups. We mention here that the class of right-ordered groups is precisely the subgroup closure of the class of lattice ordered groups (see [6], [7], [9] or [10]).

A super-reflexive Banach space is defined to be a Banach space B which has the property that no non-reflexive Banach space is finitely representable in B. Super-reflexivity is invariant under isomorphisms; a Banach space B is super-reflexive if and only if B* is super-reflexive. This concept has many equivalent formulations, some of which have been studied previously.

A connected Lie group H is generated by a pair of one-parameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups, i.e., if and only if the subalgebra generated by the corresponding pair of infinitesimal transformations is equal to the whole Lie algebra h of H (observe that the subgroup of all finite products is arcwise connected and hence, by Yamabe's theorem [5], is a sub-Lie group). If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n ; otherwise define it as infinity.

We denote by ZG the integral group ring of the finite group G. S.D. Berman [1] showed that every unit of finite order μ in G is trivial (i.e., μ = ±g for some g in G) if and only if either G is abelian or G is a Hamiltonian 2-group. In this note, we give a new and shorter proof for the “only if” part.

Let K be a field of characteristic p > 0 and let A be a separably generated algebraic function field of one variable with K as its exact constant field. Throughout this paper we shall use the following notations to classify differentials of A/K:

D(A) : the K-module of all differentials,

G(A) : the K-module of all differentials of the first kind,

R(A) : the K-module of all residue free differentials in the sense of Chevalley [2, p. 48],

E*(A) : the K-module of all pseudo-exact differentials in the sense of Lamprecht [7, p. 363], (compare the definition with our Lemma 8).

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains (see Whyburn [10], Baur [3], Krolevec [8], Dickman [5], Franklin and Kohli [6]). It was shown in [6] that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. In [6] and [7] we also related the topological properties of the domain and range of the mapping with the new domain; also these results were then used to obtain analogues and improvements of recent theorems of Arhangelskii, Čoban, Hodel, Keesling, Nagami, Okuyama, and Proizvolov. In this note we give a method of unifying the domain and range of a mapping so as to yield a meaningful open extension.

We shall call a linear function on the elements of a ring R skew-symmetric if it vanishes whenever at least two of the variables are equal. Here we shall study rings R of characteristic not 2 which satisfy the following two identities:

1

(2)is skew-symmetric.

Both of these identities hold in alternative rings. The fact that F(w, x, y, z) is skew-symmetric in alternative rings is an important tool in the study of such rings. It is also obvious that both identities hold in commutative rings. But unlike other recent generalizations of commutative and alternative rings it turns out that there exist simple, finite dimensional algebras of degree two which are neither alternative nor commutative and satisfy (1) and (2).

In this note we prove some results on the intersection properties of maximal subfields of division algebras which are finite dimensional over their centers. These results indicate that we can get very small intersections with any subalgebra if we use the appropriate maximal subfields. As a consequence of our first theorem, we obtain some theorems which are known and some which can be obtained from these known theorems (see, for instance, Theorem 3, Chapter VII in [3]). The proofs of these known results given here are very elementary and are quite different from the ones in the literature.

In a recent paper [2], the author, with F. M. Cholewinski, derived criteria for the series expansions of solutions u(x, t) of the Laguerre differential heat equation xuxx + (α + 1 - x)ux = ut in terms of the Laguerre heat polynomials and of their temperature transforms. Our present goal is the characterization of those solutions which are representable in a Maclaurin double series in xe-t and in 1 — e-t Some of the results are analogous to those derived by D. V. Widder in [4] for the classical heat equation and by the author in [1] for the generalized heat equation.

It is well-known that the space L1(Rn) of integrable functions on Euclidean space fails to be preserved by singular integral operators. As a result the rather large Lp theory of partial differential equations also fails for p = 1. Since L1 is such a natural space, many substitute spaces have been considered. One of the most interesting of these is the space we will denote by H1(Rn) of integrable functions whose Riesz transforms are integrable.

Cooper [1] called a complex-valued function f on the real line Rpositive definite for F, where F is a set of complex-valued functions on R, if the integral

exists as a Lebesgue integral and is nonnegative for every ϕ in F. Let us denote by P(F) the set of functions positive definite for F, by the set of functions in LP(R) with compact support, and by the set of functions which are locally in LP(R), i.e., for every compact subset K of R. Cooper showed that for any p ≧ 2 and that each function in is essentially bounded and hence equal almost everywhere to an ordinary continuous positive definite function in the sense of Bochner.

We can define high order derivations of an algebra into the ground field by diagrams. Then consider the same diagrams in t he category of coalgebras. By reversing all t he arrows in these diagrams, we come to a new notion - high order Lie-like elements of a coalgebra. These elements are useful in the study of the structure of coalgebras and sequences of divided powers.

In 1952, E. Hewitt and K. Yosida [3] proved that a bounded, finitely additive real-valued set function has a unique representation as the sum of a countably additive function and a “purely finitely additive” function.

Below, using a variation of the Carathéodory process we give a suitable generalization to s-bounded vector-valued set functions. In fact, since the methods do not rely on scalar multiplication, we give the result for commutative Hausdorff topological groups.

In recent years a number of inequalities have appeared which involve rearrangements of vectors in Rn and of measurable functions on a finite measure space. These inequalities are not only interesting in themselves, but also are important in investigations involving rearrangement invariant Banach function spaces and interpolation theorems for these spaces [2; 8; 9].

An O*-group is a group wherein every partial order can be extended to some full order.

THEOREM. Suppose the group G has a normal chain G = G1 ⊇ G2 ⊇ … such that

and each G/Gi is locally nilpotent and torsion-free. Then G can be embedded in thecomplete direct product G’ of divisible O*-groups.

Throughout this paper R and D will denote integral domains with the same quotient field K. A set of integral domains {Di} i∊I with quotient field K will be said to have FC (“finite character” or “finiteness condition“) if 0 ≠ ξ ∊ K implies ξ is a unit of Di for all but finitely many i. If ∩i∊IDi also has quotient field K, then {Di} has FC if and only if every non-zero element in ∩i∊IDi is a non-unit in at most finitely many Di. A non-empty set {Vi}i∊:I of rank one valuation rings with quotient field K will be called a defining family of real R-representativesfor D if {Vi} i∊:I has FC, R (⊄ ∩i∊IVi, and D = R∩ (∩i∊I Vi).

Let be an (n + p)-dimensional Kaehler manifold of constant holomorphic sectional curvature . B. O'Neill [3] proved the following result.

PROPOSITION A. Let M be an n-dimensional complex submanifold immersed in . If and if the holomorphic sectional curvature of M with respect to the induced Kaehler metric is constant, then M is totally geodesic.

He also gave the following example: There is a Kaehler imbedding of an w-dimensional complex projective space of constant holomorphic sectional curvature ½ into an -dimensional complex projective space of constant holomorphic sectional curvature 1. This shows that Proposition A is the best possible.