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Rings Of Continuous Functions [EXCLUSIVE]


As defined by Ye [\bf12], a ring is semiclean if every element is the sum of a unit and a periodic element. Ahn and Anderson [\bf1] called a ring weakly clean if every element can be written as $u+e$ or $u-e$, where $u$ is a unit and $e$ an idempotent. A weakly clean ring is semiclean. We show the existence of semiclean rings that are not weakly clean. Every semiclean ring is $2$-clean. New classes of semiclean subrings of $\r$ and $\c$ are introduced and conditions are given when these rings are clean. Cleanliness and related properties of $C(X,A)$ are studied when $A$ is a dense semiclean subring of $\r$ or $\c$.




Rings of Continuous Functions


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It 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.


Let $R$ be the ring of all continuous real valued functions on the unit interval $[0,1]$ (with pointwise operations), and let $I$ be a proper ideal of $R$. Show that there exists $λ\in [0,1]$ such that $$I\subseteq M_λ= \left\f \in R\mid f(λ)=0\right\.$$


The next observation is that the intersection of two zero sets for functions $I$ is nonempty. If to the contrary $z(f)\cap z(g)=\emptyset$, then $z(f^2+g^2)=\emptyset$, but this isn't possible since $f^2+g^2\in I$. From this proof you can see that for every $f$ and $g$ in $I$, $f^2+g^2$ is another element of $I$ whose zero set is just $z(f)\cap z(g)$. By induction, the intersection of finitely many zero sets is again a nonempty zero set of a function in $I$.


My professor briefly touched on Noetherian rings so it is still a little bit confusing. How do I go about finding these ideals? Or should I show that every ideal is generated by finitely generated elements? Any help is much appreciated!


Let X be a completely regular Hausdorff space and, as usual, C(X) be the ringof continuous functions mapping X into the line. Recall that every ring (bywhich I mean "commutative ring with unity") A can be embedded as a subring of aBaer ring B(A) - the latter being a ring in which ever annihilator of an element isprincipal generated by an idempotent. My question is: Is there a known charaterizationof spaces X such that the "Baer envelope" B((CX)) is induced by some (necessarily)dense subspace inclusion Y -> X? Basically disconnected spaces seem to havethis property. Are there others?


The small Fine-Gillman-Lambek extension generated by the classical ring of quotients, and the Riemann extension generated by Riemann $\mu$-integrable functions are both characterized as divisible envelopes of the same type of the ring of all bounded continuous functions on the Aleksandrov space. This shows the similarity of these extensions that are rather different by their origin.


ARis the sum of a unit and a projection that commute with each other. Inthis paper, we explore strong -cleanness of rings of continuous functions overspectrum spaces. We prove that a -ring R is strongly -clean if and only if Ris an abelian exchange ring and C(X) C (X) is-clean, if and only if R isan abelian exchange ring and the classical ring of quotients q(C(X)) of C(X)is -clean, where X is a spectrum space of R 041b061a72


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