Let R = ⊕i ≧0Ri be a graded integral domain, and let p ∈ Proj (R) be a homogeneous, relevant prime ideal. Let R(P) = {r/t| r ∈ Ri, t ∈ Ri\p} be the geometric local ring at p and let Rp = {r/t| r ∈ R, t ∈ R\p} be the arithmetic local ring at p. Under the mild restriction that there exists an element r1 ∈ R1\p, W. E. Kuan [2], Theorem 2, showed that r1 is transcendental over R(P) and

where S is the multiplicative system R\p. It is also demonstrated in [2] that R(P) is normal (regular) if and only if Rp is normal (regular). By looking more closely at the relationship between R(P) and R(P), we extend this result to Cohen-Macaulay (abbreviated C M.) and Gorenstein rings.

Let R be a ring with identity, Mn(R) the ring of n × n matrices over R. The lattice of two-sided ideals of R is carried via A → Mn(A) to form the lattice of two-sided ideals of Mn(R). We wish to study the more complex left ideal structure of Mn(R). For example, if K is a commutative field, then Mn(K) has non-trivial left ideals. In particular Mn(K) has the maximal left ideal consisting of all matrices with some designated column zero. Or for any ring with maximal left ideal M, Mn(R) has the maximal left ideal consisting of all matrices with some column's entries from M. In Theorem 1.2 we characterize the maximal left ideals of Mn(R) in terms of those of R. We briefly study some contraction properties of maximal left ideals in matrix rings. For R commutative we “count” the maximal left ideals of Mn(R) and describe the idealizer of any such ideal; in the case where K is a field we see that the collection of maximal left ideals of Mn(K) can be naturally identified with Pn–1(K) (projective space).

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].

Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

Heine [7, p. 93] gave the following representation for the Legendre Polynomial {Pn(x)}∞n=o

where fo,n = 1 and

Szegö [7, p. 96] generalized this result to the Ultraspherical Polynomial set ﹛Cnλ(x)﹜∞n=o and obtained

where

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such that

for all x1, …, xm in U.

Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizer

is a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

In this paper we wish to study fields which can be written as intersections of real closed fields. Several more restrictive classes of fields have received careful study (real closed fields by Artin and Schreier, hereditarily euclidean fields by Prestel and Ziegler [8], hereditarily Pythagorean fields by Becker [1]), with this more general class of fields sometimes mentioned in passing. We shall give several characterizations of this class in the next two sections. In § 2 we will be concerned with Gal , the Galois group of an algebraic closure F over F. We also relate the fields to the existence of multiplier sequences; these are infinite sequences of elements from the field which have nice properties with respect to certain sets of polynomials. For the real numbers, they are related to entire functions; generalizations can be found in [3].

In this paper a new type of inclusion theorem concerning Abel, Borel and Lambert summability is established. To state our results we need some definitions and notations. With a formal series ∑k=0∞ak, ak∈ C, and its partial sums snwe associate the series

Then ∑k=0∞ ak is said to be summable to the value s

(a) by Abel's method, if (1.1) is convergent for |v| > 1 and limv→1+A(v)= s,

(b) by Lambert's method, if (1.2) is convergent for |v| > 1 and limv→1+L(v)= s,

(c) by Borel's method, if (1.3) is convergent for all x ∈ R and limx→+∞B(x)= s,

In [6] Walter Taylor investigated the relationship between the algebraic structure of a topological algebra A and the group structure of its fundamental group π1(A) and of the higher homotopy groups πn(A),n > 1. The main result is that a variety satisfies a group law λ in homotopy (that is, π1) if and only if every group in the idempotent reduct of obeys λ. (The relevant definitions are in [6] and also § 2 of this paper.) A similar result is stated for the higher homotopy groups. As Taylor points out in the introduction, the hard part of the theorem is constructing a topological algebra in whose fundamental group may fail to obey λ; indeed, in [6] this is only done in detail for the commutative law, and the proof is rather computational.

Ramanujan's well known trigonometrical sum C(m, n) denned by

where x runs through a reduced residue system (mod n), had been shown to occur in analytic problems concerning modular functions of one variable, by Poincaré [4]. Ramanujan, independently later, used these trigonometrical sums in his remarkable work on representation of integers as sums of squares [6]. There are various generalizations of C(m, n) in the literature (some also to algebraic number fields); see, for example, [9] which gives references to some of these. Perhaps the earliest generalization to algebraic number fields is due to H. Rademacher [5]. We here consider a novel generalization involving matrices.

A remote point of a metric space X is a point in βX |Xnot in the βX-closure of any discrete subset of X. Remote points have been studied by Fine and Gillman [2], Plank [4], Robinson [6], Woods [9, 10], Van Douwen [7] and others. The main results concerning the existence of remote points are listed in Section 1. In this paper we determine some structural properties of the set of remote points of a metric space which has no isolated points. The notation of Gillman and Jerison [3] and Walker [8] will be used. All spaces are assumed to be Tychonoff.

Let X be a space. The general remote points of X, denoted TX, are those points in βX |X which are not in the βX-closure of any nowhere dense sets of X.

There are relatively few classes of local rings (R, m) for which the question of the rationality of the Poincaré series

where k = R/m, has been settled. (For an example of a local ring with non-rational Poincaré series see the recent paper by D. Anick, “Construction of loop spaces and local rings whose Poincaré—Betti series are nonrational”, C. R. Acad. Sc. Paris 290 (1980), 729-732.) In this note, we compute the Poincaré series of a certain family of local Cohen-Macaulay rings and obtain, as a corollary, the rationality of the Poincaré series of d-dimensional local Gorenstein rings (R, m) of embedding dimension at least e + d – 3, where e is the multiplicity of R. It follows that local Gorenstein rings of multiplicity at most five have rational Poincaré series.

Throughout this paper all rings considered will be commutative, noetherian with identity. If R is such a ring and M is a finitely generated R-module, we shall use v(M) to denote that non-negative integer with the property that M can be generated by v(M) elements but not by fewer.

Since every ideal in a noetherian ring is finitely generated, it is a natural question to ask what v(I) is for a given ideal I. Hilbert's Nullstellensatz may be viewed as the first general theorem dealing with this question, answering it when I is a maximal ideal in a polynomial ring over an algebraically closed field.

More recently, it has been noticed that the properties of an R-ideal I are intertwined with those of the R-module I/I2.

The objective of this paper is to extend the recent results [7, 8, 9] concerning the self-adjointness of Schrödinger-type operators with singular potentials to a more general setting. We shall be concerned here with formally symmetric elliptic differential expressions of the form

1.1

where x = (x1, …, xm) ∈ Rm (and m ≧ 1), i = (–1)1/2, ∂j = ∂/∂xj, and the coefficients ajk, bj and q are real-valued and measurable on Rm.

The basic problem that we consider is that of deciding whether or not the formal operator defined by (1.1) determines a unique self-adjoint operator in the space L2(Rm) of (equivalence classes of) square integrable complex-valued functions on Rm.

The object of this paper is to characterize the largest class of autonomous linear hereditary differential systems which generates a strongly continuous semigroup of class C0 on the product space Mp = Rn × Lp(-h, 0), 1 ≦ p < ∞, 0 < h ≦ + ∞ (R is the field of real numbers and Lp(– h, 0) is the space of equivalence classes of Lebesgue measurable maps x:[ – h, 0] ⌒ R → Rn which are p-integrable in [ –h, 0] ⌒R.) Our results extend and complete those of [4] and [15], [16] for linear hereditary differential equations possessing “finite memory” (h < + ∞ ) and those of [14], [5] and [6] in the “infinite memory case (h = + ∞ )”.

Consider the autonomous linear hereditary differential equation

(1.1)

where x(t) ∊ Rn, x:[–h, 0] ⌒ R → Rn is defined as xt(θ) = x(t + θ), C(–h, 0) is the space of bounded continuous functions [–h, 0] ⌒ R → Rn and L:C(–h, 0) → Rn is a continuous linear map.

If r is a mapping of a set X onto a set Y, then a selection for r is a mapping S of Y into X for which r o s is the identity. Analogously, if F is a mapping of a set Y into the power set 2Z of a set Z, then a selection for F is a mapping f of Y into Z such that ƒ(y) is an element of F(y) for all y in Y. These two notions are formally equivalent: Given r mapping X onto Y, define F(y) = r–l(y) and Z = X. Conversely, given F mapping Y into 2Z, define X to be the subset of Y × Z consisting of the pairs (y, z) for which z belongs to F(y), and define r on X by r(y, z) = y.

We begin with some preliminary definitions. Let S be a subset of Rd. For points x and y in S, we say x sees y via S if and only if the corresponding segment [x, y] lies in S. The set Sis said to be starshaped if and only if there is some point p in S such that, for every x in S, p sees x via S. The collection of all such points p is called the kernel of S, denoted ker S. Furthermore, if we define the star of x in S by Sx = {y: [x, y] ⊆ S}, it is clear that ker S = ⋂{Sx: x in S}.

Several interesting results indicate a relationship between ker S and the set E of (d – 2)-extreme points of S. Recall that for d ≧ 2, a point x in S is a (d – 2)-extreme point of S if and only if x is not relatively interior to a (d – 1)-dimensional simplex which lies in S. Kenelly, Hare et al. [4] have proved that if S is a compact starshaped set in Rd, d ≧ 2, then ker S = ⋂{Se: eE}. This was strengthened in papers by Stavrakas [6] and Goodey [2], and their results show that the conclusion follows whenever S is a compact set whose complement ~S is connected.

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall prove

THEOREM. If RG ≌ RH as R-algebras, then G ≌ H

The question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.

In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules

Let RG denote the group ring of a group G over a commutative ring R with unity. We recall that a group is said to be an FC-group if all its conjugacy classes are finite.

In [6], S. K. Sehgal and H. Zassenhaus gave necessary and sufficient conditions for U(RG) to be an FC-group when R is either Z, the ring of rational integers, or a field of characteristic 0.

One of the authors considered this problem for group rings over infinite fields of characteristic p ≠ 2 in [5] and G. Cliffs and S. K. Sehgal [1] completed the study for arbitrary fields. Also, group rings of finite groups over commutative rings containing Z(p), a localization of Z over a prime ideal (p) were studied in [4].

A well known theorem of Sierpiński states that every compact connected Hausdorff space is σ-connected. Hence, if X is locally compact and Hausdorff and X is locally connected at x, then x has a σ-connected neighborhood. However, local connectedness at x is not a necessary condition for x to have a σ-connected neighborhood, because the whole space may be σ-connected without being locally connected at x. One of the purposes of the present paper is then to investigate which points of a given locally compact Hausdorff space have σ-connected neighborhoods. We find also sufficient conditions for a connected, hereditarily Baire space to be σ-connected and prove the impossibility of expressing a connected, Čech-complete, rim compact space as a countable infinite union of mutually disjoint compact sets. Finally, we introduce the concept of D-connected space and relate it to σ-connectedness.

Spreads of curves were introduced by Grunbaum in [1]. A spread of curves is a continuous family of simple arcs in the real plane, every two of which intersect in exactly one point. A spread is the continuous analogue of a finite arrangement of pseudolines in the plane. Sylvester's problem for finite arrangements of pseudolines asks if every non-trivial arrangement has a simple vertex, that is a point contained in exactly two pseudolines of the arrangement. This question was answered in the affirmative by Kelly and Rottenberg [5]. One interesting feature of this result is that it does not depend on the pseudolines being straight lines.

Here we settle Sylvester's problem for spreads. We show that every nontrivial spread of line segments has uncountably many simple vertices. But we also give examples of non-trivial spreads with no simple vertices.