Ring Of Continuous Real-valued Functions Research Articles

When C(X) is an h-local ring

An h-local domain is a domain for which each nonzero prime ideal is contained in a unique maximal ideal and each nonzero element has finite character. In his dissertation, A. Omairi generalizes the notion of an h-local domain to rings with zero-divisors by restricting the definition to regular ideals. In this article, we give a characterization of when C(X) the ring of continuous real-valued functions on a space X is an h-local ring. We are left with more questions than answers.

Quasi SV-frames and their properties

Let P be a prime ideal of the ring of continuous real-valued functions on a completely regular frame L, i.e., L. We study many new results about the residue class domains L/P with an emphasis on determining when the ordered L/P is a valuation domain (i.e., when given any two non-zero elements of L/P , one divides the other). A prime ideal P of L is called a valuation prime ideal if L/P is a valuation domain. A frame L is called an SV-frame if every prime ideal of L is a valuation prime ideal. We introduce and study two new generalizations of the SV-frames. The first is that of a quasi SV-frame in which every real maximal ideal of L that is not a minimal prime ideal contains a non-maximal prime ideal P such that L/P is a valuation domain. In the second, we define a frame L to be an almost SV-frame if every maximal ideal of L contains a minimal valuation prime ideal. A point I ∈ Pt(βL) is called a special βF -point if O I = {δ ∈ L : coz δ ∈ I} is a valuation prime ideal of L. It is shown that I is a special βF -point if and only if the pseudo-prime ideals of L containing O I that are not primary form a chain under set inclusion.

Approximation by pointwise bounded sets of continuous functions

Assuming that X is a Tychonoff space, we obtain necessary and sufficient conditions for a function f∈RX to belong to the closure in RX of a pointwise bounded family F in C(X), the ring of real-valued continuous functions on X. In other words, if M(X) stands for the bidual of the space Cp(X), equipped with the pointwise topology, we provide necessary and sufficient conditions for a function f∈RX to belong to M(X). As a noteworthy fact, if M, S and R are, respectively, the Michael, the Sorgenfrey, and the real line, it is shown that C(M)⊈M(R) but C(S)⊆M(R).

New Directions in Duality Theory for Modal Logic

AbstractIn this work we present some new contributions towards two different directions in the study of modal logic. First we employ tense logics to provide a temporal interpretation of intuitionistic quantifiers as “always in the future” and “sometime in the past.” This is achieved by modifying the Gödel translation and resolves an asymmetry between the standard interpretation of intuitionistic quantifiers.Then we generalize the classic Gelfand–Naimark–Stone duality between compact Hausdorff spaces and uniformly complete bounded archimedean $\ell $ -algebras to a duality encompassing compact Hausdorff spaces with continuous relations. This leads to the notion of modal operators on bounded archimedean $\ell $ -algebras and in particular on rings of continuous real-valued functions on compact Hausdorff spaces. This new duality is also a generalization of the classic Jónsson-Tarski duality in modal logic.Abstract taken directly from the thesis.E-mail: lcarai@unisa.itURL: https://www.proquest.com/openview/5d284dbfb954383da9364149fa312b6f/1?pq-origsite=gscholar&cbl=18750&diss=y

MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

AbstractIt is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

Closed ideals in the uniform topology on the ring of real-valued continuous functions on a frame

For a completely regular frame $L$, the ring $\mathcal RL$ of real-valued continuous functions on $L$ is equipped with the uniform topology. The closed ideals of $\mathcal RL$ in this topology are studied, and a new, merely algebraic characterization of these ideals is given. This result is used to describe the real ideals of $\mathcal RL$, and to characterize pseudocompact frames and Lindelöf frames. It is shown that a frame $L$ is finite if and only if every ideal of $\mathcal RL$ is closed. Finally, we prove that every closed ideal in $\mathcal RL$ is an intersection of maximal ideals.

On the socle of a commutative ring and Zariski topology

This paper concerns the coincidence of the socle of a semiprimitive ring (or just a reduced one, in some cases) with the intersection of all essential prime, essential minimal prime, or essential maximal ideals of the ring. More precisely, we prove first that the socle of a reduced ring coincides with the intersection of all essential minimal prime ideals if and only if every minimal prime (or, equivalently, every prime) ideal is either essential or it is a direct summand which is also a maximal ideal. Next, we show that the socle of a semiprimitive ring R is equal to the intersection of all essential maximal ideals of R (i.e., Soc max ( R ) ) if and only if the set of isolated points of Max ( R ) with the Zariski topology contains no infinite basic open set. Whenever R is a semiprimitive c.a.c. ring, we prove that for every essential ideal I of R containing Soc max ( R ) , I ∕ Soc max ( R ) is essential in R ∕ Soc max ( R ) if and only if the set of isolated points of Max ( R ) is finite. We apply this result to rings of continuous real-valued functions on a topological space.

On the maximal regular ideal of pointfree function rings, and more

Let L be a completely regular frame and, as customary, let RL denote the ring of continuous real-valued functions on L. In the first part of the paper we characterize the maximal regular ideal of RL. We show that it consists precisely of the functions α such that the open sublocale of L associated with coz α is clopen and is a P-frame. We also give a characterization of this ideal in terms of the notion of the localic remainder. In contrast, the maximal regular ideal of the ring ZL of continuous integer-valued functions on L is shown to be always the zero ideal. Purity and regularity of ideals are germane in the description and characterization of the maximal regular ideal of RL. It is for this reason that in the last part of the paper we consider ideals of ideals of RL because, as we prove, it is precisely the pure ideals of RL all of whose ideals are ideals in the whole ring.

Natural extension of EF-spaces and EZ-spaces to the pointfree context

Abstract Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.

Further thoughts on the ring $${\mathcal {R}}_c (L)$$ in frames

Let C(X) denote the ring of all real-valued continuous functions on a topological space X and $${\mathcal {R}} (L)$$ be as the pointfree topology version of C(X), i.e., the ring of real-valued continuous functions on a frame L. The ring $${\mathcal {R}}_c (L)$$ is introduced as a sub-f-ring of $${\mathcal {R}} ( L)$$ as a pointfree analogue to the subring $$C_c(X)$$ of C(X) consisting of elements with the countable image. In this paper, we will study the concept of pointfree countable image in a way which will enable us to study the ring $${\mathcal {R}}_c (L)$$. In order to do so we introduce the set $$R_{\alpha } := \{ r \in {\mathbb {R}} : {{\,\mathrm{coz}\,}}(\alpha - \mathbf{r}) \not = \top \} $$ for every $$\alpha \in {\mathcal {R}} (L)$$. We prove that $$R_{\alpha } $$ is a countable subset of $${\mathbb {R}}$$ for every $$\alpha \in {\mathcal {R}}_c (L)$$. Next, we show that if L is a compact frame, then $$R_{\alpha } $$ is a finite subset of $${\mathbb {R}}$$ for every $$\alpha \in {\mathcal {R}}_c (L)$$. Also, we study the result which says that for any topological space X there is a zero-dimensional space Y which is a continuous image of X and $$C_c(X) \cong C_c(Y )$$ in pointfree topology. Finally, we prove that, for some frame L, the ring $${\mathcal {R}}_c (L)$$ may not be isomorphic to $${\mathcal {R}} (M)$$, for any given frame M.

Some Notes on z-Ideal and d-Ideal in $$\mathcal {R}L$$

As usual, the ring of continuous real-valued functions on a completely regular frame L is denoted by $$\mathcal {R}L$$. It is well known that an ideal Q in $$\mathcal {R}L$$ is a z-ideal if and only if $$\sqrt{Q}$$ is a z-ideal in which case $$Q=\sqrt{Q}$$. We show the same fact in d-ideal context and then it turns out that the sum of a primary ideal and a z-ideal (d-ideal) in $$\mathcal {R}L$$ which are not in a chain is a prime z-ideal (d-ideal). For an ideal (a non-regular ideal) Q in $$\mathcal {R}L$$, we characterize the greatest z-ideal (d-ideal) contained in Q and the smallest z-ideal (d-ideal) containing Q in terms of basic z-ideals (d-ideals). We characterize frames L for which prime z-ideals and d-ideals coincide in $$\mathcal {R}L$$, or equivalently the sum of any two ideals consisting entirely of zero-divisors consists entirely of zero-divisors. Some characterizations of the quasi F-frame, extremally disconnected frames and P-frames in terms of d-ideals are provided. Finally, we show that a reduced ring R is regular if and only if every prime d-ideal in R is maximal.

Annihilator-stability and unique generation in C(X)

Using the equivalence of unique generation and cleanness of [Formula: see text], we give affirmative answers to questions raised in [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464–491] and [D.D. Anderson et al., When are associates unit multiples? Rocky Mountain J. Math. 34 (2004) 811–828] for rings of real-valued continuous functions. In fact, we show that if [Formula: see text] is [Formula: see text] (uniquely generated) then [Formula: see text] is too, and [Formula: see text] is strongly associate if and only if [Formula: see text] is, where [Formula: see text]. We give topological characterizations of [Formula: see text] and [Formula: see text] (annihilator-stable) elements of [Formula: see text] for continuum spaces [Formula: see text] and using this, we observe that the product of two UG elements need not be [Formula: see text]. It is shown that the set of elements in [Formula: see text] which have stable range 1 and the set of [Formula: see text] elements of [Formula: see text] coincide and several examples are given which show that the set of [Formula: see text] elements, the set of [Formula: see text] elements and the set of clean elements of [Formula: see text] can differ. Finally, we characterize spaces [Formula: see text] for which every clean element of [Formula: see text] or every element of [Formula: see text] which has stable range 1 is [Formula: see text].

Concerning real-closed ideals in [formula omitted] and SV-frames

Let RL be the ring of continuous real-valued functions on a completely regular frame L. We study the class of prime ideals P of the ring RL determined by the condition: RL/P is a real-closed ring. We give some necessary and sufficient frame-theoretic conditions for an ordered integral domain of the form RL/P to be a real-closed ring, when P is a prime z-ideal (d-ideal) of RL. A completely regular frame L is called an SV-frame if for every prime ideal P of RL, the ordered integral domain RL/P is a real-closed ring. It is shown that every C⁎-quotient of an SV-frame is an SV-frame. We also show that open quotients ↓c in an SV-frame are SV-frames for all cozero elements c. Larson [22] has given a topological characterization of compact SV-spaces. By extending this characterization to frames, we show that the compactness limitation can really be relaxed, even in spaces, and so strengthen Larson's result.

Decision procedures for the conditions true in certain metric structures

After establishing a completeness theorem for continuous logic, Ben Yaacov and Pedersen conclude that if T is a complete recursive L-theory in continuous logic, and v(φ) is the truth value of the L-sentence φ in models of T, then v(φ) is a recursive real uniformly recursive in φ. Some of the examples to which the latter result applies are theories of the following structures: atomless probability structures, the Urysohn space of diameter 1, Hilbert space, the lattice-ordered group or ring of real-valued continuous functions on the Cantor set, and the complex ⁎-algebra of continuous functions on the Cantor set. This paper will explain why these examples obey much stronger results, yielding (for example) decision procedures for the conditions true in these structures.

On injectivity of the ring of real-valued continuous functions on a frame

We give characterizations of $P$-frames and extremally disconnected $P$-frames based on ring-theoretic features of the ring of continuous real- valued functions on a frame $L$, i.e. $\mathcal RL$. It is shown that $L$ is a $P$-frame if and only if $\mathcal RL$ is an $\aleph_0$-self-injective ring. Consequently for pseudocompact frames if $\mathcal RL$ is $\aleph_0$-self-injective, then $L$ is finite. We also prove that $L$ is an extremally disconnected $P$-frame iff $\mathcal{R}L$ is a self-injective ring iff $\mathcal{R}L$ is a Baer regular ring iff $\mathcal{R}L$ is a continuous regular ring iff $\mathcal{R}L$ is a complete regular ring.

Pointfree topology version of image of real-valued continuous functions

Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree version of $C_c(X).$The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in $ {mathcal{R}} L$. In particular, we will introduce the pointfree version of the ring $C_c(X)$. We define a relation from $ {mathcal{R}} L$ into the power set of $mathbb R$, namely overlap . Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as $mathop{hbox{Im}} (f) subseteq S$ for every continuous function $f:Xrightarrowmathbb R$ and $ S subseteq mathbb R$.

On quasi-normality of function rings

An $f$-ring is called quasi-normal if the sum of any two different minimal prime $\ell $-ideals is either a maximal $\ell $-ideal or the entire $f$-ring. Recall that the \textit {zero-component} of a prime ideal $P$ of a commutative ring $A$ is the ideal \[ O_P=\{a\in A\mid ab=0 \mbox { for some } b\in A\setminus P\}. \] \vspace {0.5pt} \noindent Let $C(X)$ be the $f$-ring of continuous real-valued functions on a Tychonoff space $X$. Larson proved that $C(\beta X)$ is quasi-normal precisely when $C(X)$ is quasi-normal and the zero-component of every hyper-real ideal of $C(X)$ is prime. We show that this result is actually purely ring-theoretic and thus deduce its extension to the $f$-rings $\mathcal {R}L$ of continuous real-valued functions on a frame $L$. A subspace of $X$ is called a $2$-boundary subspace if it is of the form $cl _X(C)\cap cl _X(D)$ for some disjoint cozero-sets $C$ and $D$ of $X$. For normal spaces, Kimber proved that $C(X)$ is quasi-normal precisely when every $2$-boundary subspace of $X$ is a $P$-space. By viewing spaces as locales, we obtain a characterization along similar lines which does not require normality, namely, for any Tychonoff space $X$, $C(X)$ is quasi-normal if and only if every $2$-boundary sublocale of the Lindel\"{o}f reflection of $X$ in the category of locales is a $P$-frame.

On minimal ideals in the ring of real-valued continuous functions on a frame

On minimal ideals in the ring of real-valued continuous functions on a frame

Commutative rings in which zero-components of essential primes are essential

For a prime ideal [Formula: see text] of a commutative ring [Formula: see text] with identity, we denote (as usual) by [Formula: see text] its zero-component; that is, the set of members of [Formula: see text] that are annihilated by nonmembers of [Formula: see text]. We study rings in which [Formula: see text] is an essential ideal, whenever [Formula: see text] is an essential prime ideal. We characterize them in terms of the lattices (which are, in fact, complete Heyting algebras) of their radical ideals. We prove that the classical ring of quotients of any ring of this kind is itself of this kind. We show that direct products of rings of this kind are themselves of this kind. We observe that the ring of real-valued continuous functions on a Tychonoff space is of this kind precisely when the underlying set of the space is infinite. Replacing [Formula: see text] with the pure part of [Formula: see text], we obtain a formally stronger variant which is still characterizable in terms of the lattices of radical ideals.

English

Let $X$ be a completely regular Hausdorff space and, as usual, let $C(X)$ denote the ring of real-valued continuous functions on $X$. The lattice of $z$-ideals of $C(X)$ has been shown by Martinez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) $\beta X$ precisely when $X$ is a $P$-space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a $d$-ideal if whenever two elements have the same annihilator and one of the elements belongs to the ideal, then so does the other. We characterize when the spectrum of the lattice of $d$-ideals of $C(X)$ is the Stone-Cech compactification of the largest dense sublocale of the locale determined by $X$. It is precisely when the closure of every open set of $X$ is the closure of some cozero-set of $X$.