Physical Sciences and Mathematics Commons^™

Quasi F-Covers Of Tychonoff Spaces, Melvin Henriksen, J. Vermeer, R. G. Woods

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A Tychonoff topological space is called a quasi F-space if each dense cozero-set of X is C*-embedded in X. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi F-cover" QF(X) of a compact space X as an inverse limit space, and identify the ring C(QF(X)) as the order-Cauchy completion of the ring C*(X). In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi F-cover of an arbitrary Tychonoff space. In this paper the minimal quasi F-cover of a compact space X is constructed as the space of ultrafilters …

Some Properties Of Positive Derivations On F-Rings, Melvin Henriksen, Frank A. Smith

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Throughout A denotes an f-ring; that is, a lattice-ordered ring that is a subdirect union of totally ordered rings. We let D(A) denote the set of derivations D: A --> A such that a ≥ 0 implies Da ≥ 0, and we call such derivations positive. In [CDK], P. Coleville, G. Davis, and K. Keimel initiated a study of positive derivations on f-rings. Their main results are (i) D ε D(A) and A archimedean imply D = 0, and (ii) if A has an identity element 1 and a is the supremum of a set …

Multiplicatively Periodic Rings, Ted Chinburg, Melvin Henriksen

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We prove a generalization of Luh's result without using Dirichlet's Theorem. We then use Theorem 1 to show that the J-subrings of a periodic ring form a lattice with respect to join and intersection (the join of two subrings is the smallest subring containing both of them). After noting that every J-ring has nonzero characteristic, we determine for which positive integers n and m there exist J-rings of period n and characteristic m. This generalizes a problem posed by G. Wene.

A Simple Characterization Of Commutative Rings Without Maximal Ideals, Melvin Henriksen

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In a course in abstract algebra in which the instructor presents a proof that each ideal in a ring with identity is contained in a maximal ideal, it is customary to give an example of a ring without maximal ideals.

Sums Of Kth Powers In The Ring Of Polynomials With Integer Coefficients, Ted Chinburg, Melvin Henriksen

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A working through of two theorems.

Suppose R is a ring with identity element and k is a positive integer. Let J(k, R) denote the subring of R generated by its kth powers. If Z denotes the ring of integers, then G(k, R) = {a ∈ Z: aR ⊂ J(k, R)} is an ideal of Z.

The Space Of Minimal Prime Ideals Of A Commutative Ring, Melvin Henriksen, Meyer Jerison

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The present paper is devoted to the space of minimal prime ideals of a more-or-less arbitrary commutative ring. Rings C(X) of continuous functions on topological spaces X appear only in §5 where they serve largely to provide significant examples.

On Rings Of Bounded Continuous Functions With Values In A Division Ring, Ellen Correl, Melvin Henriksen

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Let C*(X, A) denote the ring of bounded continuous functions on a (Hausdorff) topological space X with values in a topological division ring A. If, for every maximal (two-sided) ideal M of C*(X, A), we have C*(X, A)/M is isomorphic with A, we say that Stone's theorem holds for C*(X, A). It is well known [9; 6] that Stone's theorem holds for C*(X, A) if A is locally compact and connected, or a finite field. In giving a partial answer to a question of Kaplansky [7], Goldhaber and Wolk showed in [5] that, with restriction on X, and if A …

Some Remarks About Elementary Divisor Rings, Leonard Gillman, Melvin Henriksen

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By a slight modification of Kaplansky's argument, we find that the condition on zero-divisors can be replaced by the hypothesis that S be an Hermite ring (i.e., every matrix over S can be reduced to triangular form). This is an improvement, since, in any case, it is necessary that S be an Hermite ring, while, on the other hand, it is not necessary that all zero-divisors be in the radical. In fact, we show that every regular commutative ring with identity is adequate. However, the condition that S be adequate is not necessary either.

We succeed in obtaining a necessary …

Rings Of Continuous Functions In Which Every Finitely Generated Ideal Is Principal, Leonard Gillman, Melvin Henriksen

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The outline of our present paper is as follows. In §1, we collect some preliminary definitions and results. §2 inaugurates the study of F-rings and F-spaces (i.e., those spaces X for which C(X) is an F-ring).

The space of reals is not an F-space; in fact, a metric space is an F-space if and only if it is discrete. On the other hand, if X is any locally compact, σ-compact space (e.g., the reals), then βX-X is an F-space. Examples of necessary and sufficient conditions for an arbitrary completely regular space to be an F-space are:

(i) for every f …

Some Remarks On Elementary Divisor Rings Ii, Melvin Henriksen

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A commutative ring S with identity element 1 is called an elementary divisor ring (resp. Hermite ring) if for every matrix A over S there exist nonsingular matrices P, Q such that PAQ (resp. AQ) is a diagonal matrix (resp. triangular matrix). It is clear that every elementary divisor ring is an Hermite ring, and that every Hermite ring is an F-ring (that is, a commutative ring with identity in which all finitely generated ideals are principal).

On A Theorem Of Gelfand And Kolmogoroff Concerning Maximal Ideals In Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen, Meyer Jerison

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This paper deals with a theorem of Gelfand and Kolmogoroff concerning the ring C= C(X, R) of all continuous real-valued functions on a completely regular topological space X, and the subring C* = C*(X, R) consisting of all bounded functions in C. The theorem in question yields a one-one correspondence between the maximal ideals of C and those of C*; it is stated without proof in [2]. Here we supply a proof (§2), and we apply the theorem to three problems previously considered by Hewitt in [5].

Our first result (§3) consists of two simple constructions of the Q-space vX. …

On The Prime Ideals Of The Ring Of Entire Functions, Melvin Henriksen

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Let R be the ring of entire functions, and let K be the complex field. In an earlier paper [6], the author investigated the ideal structure of R, particular attention being paid to the maximal ideals. In 1946, Schilling [9, Lemma 5] stated that every prime ideal of R is maximal. Recently, I. Kaplansky pointed out to the author (in conversation) that this statement is false, and constructed a non maximal prime ideal of R (see Theorem 1(a), below). The purpose of the present paper is to investigate these nonmaximal prime ideals and their residue class fields. The author is …

On The Ideal Structure Of The Ring Of Entire Functions, Melvin Henriksen

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Let R be the ring of entire functions, and let K be the complex field. The ring R consists of all functions from K to K differentiable everywhere (in the usual sense).

The algebraic structure of the ring of entire functions seems to have been investigated extensively first by O. Helmer [1].

The ideals of R are herein classified as in [2]: an ideal I is called fixed if every function in it vanishes at at least one common point; otherwise, I is called free. The structure of the fixed ideals was determined in [1]. The structure of the …