Let R be a finite ring with 1 and let R* denote the group of units of R. Let p be a prime number. In this paper we consider the question of whether there exist a, b in R* such that a and b are non- p -th powers whose sum is 1. If such units a, b existing, we say that R is an N (p)-ring. Of course if p does not divide |R*|, the order of R*, then every element in R* is a pth power.

In this paper, we will prove, as a consequence of the main theorem,

THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.

Let be a projective plane and a subplane of . If l is a line of , we let denote the group of all elations in that have as axis and leave Q invariant. In [12, p. 921], Ostrom asked for a description of all finite planes that have a Baer subplane with the property that for all lines l of . Here denotes the order of G. Both the desarguesian planes of square order and the generalized Hughes planes have this property (Hughes [10], Ostrom [14], Dembowski [6]). One of the aims of this paper is to show that these are the only planes having such a Baer subplane.

In this paper we study conditions under which certain localizations of semilocal Hi and Di rings satisfy the second chain condition. Two of these results are as follows.

• 1) If P is a prime in a semilocal Hi ring with depth P > i ≧ height P — 1, then RP satisfies the second chain condition.

  2) If P is a prime in a semilocal Dt ring with depth P ≧ i, then RP satisfies the second chain condition.

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

Our method using CH is a blend of two earlier constructions (Hajnal-Juhász [2] and Ostaszewski [4]) of hereditarily separable (HS), regular, non-Lindelöf, first countable spaces. [4] produces a much better space than ours in § 1 ; it has all of our properties except that it is not realcompact (which is probably more interesting), and it is countably compact as well; however, the construction works only under ◇, which implies the continuum hypothesis (CH) but is not equivalent to it.

Let be an adapted sequence of integrable random variables on the probability space . Let us set .The following result can be immediately derived from Brown [2]:

Let G be an (open) domain in the finite complex plane and F a relatively closed proper subset of G. We denote by M(G) the set of functions meromorphic on G and as usual by R(K) (for a compact set K) the set of uniform limits of rational functions without poles on K.

Let Specie denote the set of prime ideals of a commutative ring with identity R, ordered by inclusion; and call a partially ordered set spectral if it is order isomorphic to Spec R for some R. What are some conditions, necessary or sufficient, for a partially ordered set X to be spectral? The most desirable answer would be the type of result that would allow one to stare at the diagram of a given X and then be able to say whether or not X is spectral. For example, it is known that finite partially ordered sets are spectral (see [2] or [5]).

A partially ordered set P has the fixed point property if every order-preserving map f : P → P has a fixed point, i.e. there exists x ∊ P such that f(x) = x. A. Tarski's classical result (see [4]), that every complete lattice has the fixed point property, is based on the following two properties of a complete lattice P:

• (A) For every order-preserving map f : P → P there exists x ∊ P such that x ≦ f(x).

• (B) Suprema of subsets of P exist; in particular, the supremum of the set {x|x ≦ f(x)} ⊂ P exists.

A C1-mapping ƒ from the oriented circle S1 into the oriented plane R2 such that f f’ (t) ≠ 0 for all t is called a regular immersion. We call a point p in Im f a double point if f-1(p) is a two element set with the corresponding tangent vectors being linearly independent. A regular immersion which is one-to-one except at a finite number of points whose images are double points is called a normal immersion. The work of Whitney [7], Titus [3] and Verhey [6] shows that the normal immersions form a dense open subset in the space of regular immersions with the usual C1-topology, and can be characterized up to diffeomorphic equivalence by a combinatorial invariant called the intersection sequence.

Let ƒ be a function meromorphic in the finite complex plane C. If for some number θ, 0 ≦ θ < 2 π, the family, fr(z) = f(reθz), is not normal at z = 1, then the ray arg z = θ is called a Julia ray. Such a ray has the property that in every sector containing it, F assumes every value infinitely often with at most two exceptions. Many authors have taken this property as the definition of a Julia ray.

A matrix H of order n = 4t with all entries from the set ﹛1, —1﹜ is Hadamard if HHt = 4tI. The set of Hadamard matrices is . A matrix is of type I or is skew-Hadamard if H = S — I where St = —S (some authors also use H = S + I). The set of type I members is . A matrix P is a signed permutation matrix if each row and each column has exactly one non-zero entry, and that entry is from the set ﹛1, —1﹜.

Throughout this paper all rings will have unity and all modules will be unital.

If X ⊂ R, then r(X) (respectively, l(X)) denotes the right (left) annihilator of x.

An element d of R is called right (left) regular if r(d) = 0 (1(d) = 0). An element which is both right and left regular is called regular.

Definition. An orthogonal design in order n and of type (u1, … , us) on the commuting variables x1, . . . , xs is an n X n matrix, X, with entries from the set ﹛0, ±x1, … , ±xs﹜ such that

The stacking method (see [1] and [5, Section 6]) has been used with great success in ergodic theory to construct a wide variety of examples of ergodic transformations (see, for example, [1 ; 3 ; 4; 5; 7]). However very little is known in general about the class of transformations which can be constructed by the stacking method using single stacks. In particular there is no simple characterization of the class .

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

Throughout this paper, the Alexander-Spanier cohomology with compact supports will be used. Suppose X is a compact connected topological ra-manifold which admits an effective action of a compact connected Lie group G (m ≧ 19).

To construct a well behaved localization of a noetherian ring R at a semiprime ideal S, it seems necessary to assume that the set (S) of modulo S regular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ring Rs in addition: one calls such 5 classical. To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

This article, as the name implies, is a continuation of [9]. In that article the second author investigates finite translation planes containing both affine dations and affine homologies. (See the beginning of Section 2 for definitions.) Such translation planes are called Ei7-planes. In [9] the author restricted himself to translation planes of characteristic p ≧ 5. The main reasons for this were that Ostrom's and Hering's theorem [13;4] on affine dations excluded the case p = 3 and the conclusions were easier to interpret geometrically when p ≧ 5 (as opposed to the case p = 2). Since then Ostrom [17] has settled the case p = 3.