Ideals of Ring of Continuous Functions
On $z$-ideals of pointfree function rings
authors
abstract
Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of continuous real-valued functions on $L$. We show that the lattice $Zid(mathcal{R}L)$ of $z$-ideals of $mathcal{R}L$ is a normal coherent Yosida frame, which extends the corresponding $C(X)$ result of Mart'{i}nez and Zenk. This we do by exhibiting $Zid(mathcal{R}L)$ as a quotient of $Rad(mathcal{R}L)$, the frame of radical ideals of $mathcal{R}L$. The saturation quotient of $Zid(mathcal{R}L)$ is shown to be isomorphic to the Stone-v{C}ech compactification of $L$. Given a morphism $hcolon Lto M$ in $mathbf{CRegFrm}$, $Zid$ creates a coherent frame homomorphism $Zid(h)colonZid(mathcal{R}L)toZid(mathcal{R}M)$ whose right adjoint maps as $(mathcal{R}h)^{-1}$, for the induced ring homomorphism $mathcal{R}hcolonmathcal{R}Ltomathcal{R}M$.Thus, $Zid(h)$ is an $s$-map, in the sense of Mart`{i}nez cite{Mar1}, precisely when $mathcal{R}(h)$ contracts maximal ideals to maximal ideals.
Upgrade to premium to download articles
Sign up to access the full text
sign up
Already have an account?login
Login
similar resources
on $z$-ideals of pointfree function rings
let $l$ be a completely regular frame and $mathcal{r}l$ be the ring of continuous real-valued functions on $l$. we show that the lattice $zid(mathcal{r}l)$ of $z$-ideals of $mathcal{r}l$ is a normal coherent yosida frame, which extends the corresponding $c(x)$ result of mart'{i}nez and zenk. this we do by exhibiting $zid(mathcal{r}l)$ as a quotient of $rad(mathcal{r}l)$, the ...
full text
Concerning the frame of minimal prime ideals of pointfree function rings
Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of continuous real-valued functions on $L$. We study the frame $mathfrak{O}(Min(mathcal{R}L))$ of minimal prime ideals of $mathcal{R}L$ in relation to $beta L$. For $Iinbeta L$, denote by $textit{textbf{O}}^I$ the ideal ${alphainmathcal{R}Lmidcozalphain I}$ of $mathcal{R}L$. We show that sending $I$ to the set of minimal prime ...
full text
On The Function Rings of Pointfree Topology
The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not o...
full text
Zero sets in pointfree topology and strongly $z$-ideals
In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...
full text
Zero elements and $z$-ideals in modified pointfree topology
In this paper, we define and study the notion of zero elements in topoframes; a topoframe is a pair $(L, tau)$, abbreviated $L_{ tau}$, consisting of a frame $L$ and a subframe $ tau $ all of whose elements are complemented elements in $L$. We show that the $f$-ring $ mathcal{R}(L_tau)$, the set of $tau$-real continuous functions on $L$, is uniformly complete. Also, t...
full text
Extending and contracting maximal ideals in the function rings of pointfree topology
By first describing the fixed maximal ideals of RL and those of its bounded part, R∗L, we show that every fixed maximal ideal of the bigger ring contracts to a fixed maximal ideal of the smaller ring, and every fixed maximal ideal of the smaller ring extends to a fixed maximal ideal of the bigger ring. However, the only instance where every maximal ideal of R∗L extends to a maximal ideal of RL ...
full text
Download full text
Sign up to access the full text
Signup
Already have an account?login
Login
Journal title:
- Bulletin of the Iranian Mathematical Society
volume 40 issue 3
pages 657- 675
publication date 2014-06-01
By following a journal you will be notified via email when a new issue of this journal is published.
Source: https://www.virascience.com/en/paper/on-z-ideals-of-pointfree-function-rings/
0 Response to "Ideals of Ring of Continuous Functions"
Post a Comment