Ring Of Continuous Real-valued Functions Research Articles

The pointwise feedback relation for linear dynamical systems

Let ∑ and ∑' be two m-input n-dimensional linear dynamical systems over a commutative ring R. If ∑ is feedback equivalent to ∑' then ∑ is pointwise feedback equivalent to ∑' (i.e. for all prime ideal p of R the systems ∑( p) and ∑'( p) are feedback equivalent over the residue field k( p), where ∑( p) is the natural extension of ∑ to k( p)). This paper is devoted to the study of the pointwise feedback relation. We characterize when two systems ∑ and ∑' are pointwise equivalent. We show that R is an absolutely flat ring if and only if the feedback relation and the pointwise feedback relation are equivalent. The particular case of rings of real-valued continuous functions is specially treated.

H-enrichments of topologies

An H-enrichment of a topology T on a set X is a topology L on X such that L ⊇ T and every homeomorphism from X to itself with respect to T is also a homeomorphism with respect to L. An H-enrichment is a C-enrichment if “homeomorphism” can be replaced by “continuous function” above. Generally in “nice” spaces, there is a scarcity of C-enrichments and an abundance of H-enrichments. We capitalize on the scarcity of C-enrichments to prove classification theorems for minimally free rings of continuous real-valued functions; with H-enrichments in general, we focus on separation and connectedness axioms.

Approximating PA

It is shown that the ring of continuous real-valued functions on a compact Hausdorff space of dimension $\leq 1$ is a BCS-ring. This result follows from a study of the preservation of the PA and BCS properties under certain ring-theoretic constructions.

Approximating PA

It is shown that the ring of continuous real-valued functions on a compact Hausdorff space of dimension ≤ 1 \leq 1 is a BCS-ring. This result follows from a study of the preservation of the PA and BCS properties under certain ring-theoretic constructions.

On the intrinsic topology and some related ideals of 𝐶(𝑋)

The above topology is defined and studied on C ( X ) C(X) , the ring of real-valued continuous functions on a completely regular Hausdorff space X X . The minimal ideals and the socle of C ( X ) C(X) are characterized via their corresponding z z -filters. We observe that these ideals are z z -ideals and X X is discrete if and only if the socle of C ( X ) C(X) is a free ideal. It is also shown that for a class of topological spaces, containing all P P -spaces, the family C k ( X ) {C_k}(X) of functions with compact support is identical with the intersection of the free maximal ideals of C ( X ) C(X) .

On productive classes of function rings

No nontrivial P P -class (" P P " for "productive") of rings of continuous real-valued functions can be category equivalent to any elementary P P -class of finitary universal algebras.

On Productive Classes of Function Rings

No nontrivial $P$-class ("$P$" for "productive") of rings of continuous real-valued functions can be category equivalent to any elementary $P$-class of finitary universal algebras.

Prime z-Ideals of C(X) and Related Rings

Let C(X) be the ring of continuous real-valued functions on a (completely regular) topological space X. The structure of the prime ideals and the prime z-ideals of C(X) has been the subject of much investigation (see e-g- [1], [3], [5]). One of the surprising facts about C(X) is that the sum of two prime ideals is again prime.

Topological Rings of Quotients

We investigate here the notion of a topological ring of quotients of a topological ring with respect to an arbitrary Gabriel (idempotent) filter of right ideals. We describe the topological ring of quotients first as a subring of the algebraic ring of quotients, and then show it is a topological bicommutator of a topological injective R-module. Unlike R. L. Johnson in [6] and F. Eckstein in [2] we do not always make the ring an open subring of its ring of quotients. This would exclude examples such as C(X), the ring of continuous real-valued functions on a compact space, and its ring of quotients as described in Fine, Gillman and Lambek [3].

Functions with pseudocompact support

Let X be a completely regular Hausdorff space, C( X) the ring of real-valued continuous functions on X, C K the ideal of functions with compact support, I the intersection of the free maximal ideals of C( X), and C ψ the ideal of functions with pseudocompact support. For any space, C K ⊆ I ⊆ C ψ . When C K = I, or I = C ψ , or C K = C ψ , it is said that X is μ-compact, η-compact, or ψ-compact, respectively. These ideals and spaces are characterized in terms of the ideal structure of C( X) and the topological structure of β X. Also, μ-, η-, and ψ-compactifications are constructed. Examples and counterexamples are given.

Submodules of 𝐶(𝑋)×⋯×𝐶(𝑋)

Let C ( X ) C(X) be the ring of continuous real-valued functions on a compact Hausdorff space X with the sup norm topology. In the present paper, the closed submodules of the C ( X ) C(X) -module C ( X ) × ⋯ × C ( X ) C(X) \times \cdots \times C(X) are characterized and a necessary and sufficient condition for a submodule of C ( X ) × ⋯ × C ( X ) C(X) \times \cdots \times C(X) to be closed and finitely generated is established.

An algebraic characterization of dimension

The purpose of this paper is to translate the condition defining Lebesgue covering dimension of a topological space X into a condition on C ( X ) C(X) , the ring of continuous real-valued functions on X.

Decomposition spectra of rings of continuous functions

Let S be a subset of a completely regular Hausdorff space X. Sufficient conditions on S and X are obtained for the ring of continuous real-valued functions on 5 to be isomorphic to an inverse limit of quotient rings of the ring of continuous functions on X, or, alternatively, of the ring of bounded continuous functions on X. An application to the theory of rings of quotients of rings of continuous functions is given.

The Dedekind completion ofC(𝒳)

The question to which this study addresses itself is the following: given a completely regular space <%f, is the Dedekind completion of G{<%f) isomorphic to C(g^) for some space ^/η. Here, C(J^) denotes the ring of continuous realvalued functions on <%f under pointwise order. Affirmative answers were provided by Dilworth for the class of compact spaces in 1950 and by Weinberg for the class of countably paracompact and normal spaces in 1960. It remained an open question whether there were any spaces for which a negative answer held. In this paper, we provide a necessary and sufficient condition that the Dedekind completion of Cί^ 7 ), for ^ a realcompact space, be isomorphic to C(^/) for some ^/. Using this, we are able to provide an example of a space gf for which the Dedekind completion of G(<%?) is not isomorphic to C{^/) for any space f. Specifically, we define and characterize a class of spaces which we call weak cb-spaces: those spaces J2f with the property that every locally bounded, lower semicontinuous function on ^ is bounded above by a continuous function. We then prove that for an arbitrary (completely regular) space J2f, the Dedekind completion of C{^f) is isomorphic to some C(^/) if and only if v^7 (the Hewitt realcompactification of g?) is a weak cδ-space. The sufficiency of this condition actually generalizes Weinberg's result, as is shown by examples; the necessity provides the negative result referred to above. The preliminary investigation of the Dedekind completion is done in Section 1, in the setting of an arbitrary (^-algebra. In Section 2, we study the connection between the lattice of normal upper semicontinuous functions on a completely regular space ^ and the minimal projective extension of gf'. This leads to the observation, in Section 4, that for a weak c6-space g? the Dedekind completion of C{^f) is isomorphic to C(3/)y where ^/ is the minimal projective extension of gf. Weak cδ-spaces are studied in Section 3, and Section 4 contains our main result. It is a pleasure to acknowledge fruitful conversations and correspondence on the subject of this paper with G. A. Jensen, J. E. Kist, and E. C. Weinberg. Their contributions and encouragement are sincerely appreciated.

Vector bundles and projective modules

Serre [9, ?50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring. For some time, it has been assumed thlat a similar correspondence exists between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring of continuous real-valued functions on X. A number of examples of projective modules have been given using this correspondence. However, no rigorous treatment of the correspondence seems to have been given. I will give such a treatment here and then give some of the examples which may be constructed in this way.

Correspondances singulières par parallélisme des plans tangents des deux surfaces

3. L. Gillman and M. Henriksen, Concerning rings of continuous functions, Trans. Amer. Math. Soc. vol. 77 (1954) pp. 340-362. 4. -, Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc. vol. 82 (1956) pp. 366-391. 5. L. Gillman, M. Henriksen and M. Jerison, On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. vol. 5 (1954) pp. 447-455. 6. E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. vol. 64 (1948) pp. 45-99. 7. J. G. Home, On the ideal structure of certain semirings and compactification of topological spaces, to appear in Trans. Amer. Math. Soc. 8. J. L. Kelley, General topology, New York, Van Nostrand, 1955. 9. N. H. McCoy, Rings and ideals, Mathematical Association of America, 1948. 10. A. N. Milgram, Multiplicative semigroups ofcontinuous functions, Duke Math. J. vol. 16 (1949) pp. 377-383.

Compact sets of functions and function rings

A widely used theorem of analysis asserts that a uniformly bounded, equicontinuous family of functions has a compact closure in the space of continuous functions. This lemma, variously attributed to Arzela, Escoli, Montel, Vitali, and so on, is of importance in the theory of integral equations, conformal mapping, calculus of variations, and so on. In recent years the lemma has been generalized by S. B. Myers [1 ].I A part of his results may be formulated as follows; If a topological space X is either (a) locally compact, (b) satisfies the first axiom of countability, and if Y is a metric space, then a family F of continuous functions from X to Y is compact (in a suitable topology) if and only if (1) F(x) = UfEFf(x) is compact for all xCX, (2) F is closed, (3) F is equicontinuous. The main purpose of ?1 of this paper is to characterize compact sets of functions when Y is any regular topological space. The problem is therefore to find a condition to replace equicontinuity, which no longer makes sense. We obtain such a characterization which holds for an easily described class of spaces X which includes both locally compact and first countable. In ?2 these results are applied to obtain a sort of duality theorem for the ring of real-valued continuous functions, R(X), on a space X. Namely, it is shown that, under quite general conditions, the space X is homeomorphic with the space H(X) of continuous homomorphisms from the ring R(X) onto the real numbers, where R(X) and H(X) are given the compact-open topology.

Rings of Real-Valued Continuous Functions. I

Rings of Real-Valued Continuous Functions. I

Rings of real-valued continuous functions. I

The study, by the last-named author, of the ring of all real-valued continuous functions ~(X, IR) on a tpopological space X, was begun some 33 years ago in the article [1]. The principal notions of this paper are realcompact spaces (originally named Q-spaces for no very good reason) and hyperreal fields (which we call here H-fields). These concepts have found applications in a number of areas. First, topologists have found realcompact spaces to be of interest: see for example [16] and [19]. Second, ultraproducts and ultrapowers, first considered by Log [31], have proved to be a powerful model-theoretic tool for investigation from a unified point of view of varied problems in set theory, logic, model theory, certain branches of algebra, and the theory of numbers (Artin's hypothesis and p-adic fields): see [40] and [8]. The method of ultraproducts and ultrapowers has been worked out in detail, and one may say that a more or less complete theory exists. Its traditional aspects are set forth in the two monographs of Chang and Keisler ([7] and [8]), in the monograph of Comfort and Negrepontis [14], and in the monograph of A.I. Mal'cev [33]. In particular, much progress has been made in such classical problems as the structure of ultrafilters and the cardinal numbers of ultraproducts. The key to the results obtained up to now lies in axiomatic set theory, generic models, and the theory of large cardinal numbers. Studies in recent years have tended to neglect the theory of H-fields. One should not forget, however, that ultrapowers of 1R are exactly the H-fields obtained from discrete spaces. Hence the study of arbitrary H-fields, their algebraic structure, classification, and isomorphism types, should and does lead to more complicated topological and set-theoretical problems. Little progress has been made up to now with