Extending and contracting maximal ideals in the function rings of pointfree topology
نویسنده
چکیده
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 is when the two rings coincide. This allows the deduction that RL is integral over R∗L if and only if L is pseudocompact.
منابع مشابه
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 ...
متن کاملZero 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),...
متن کاملConcerning 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 ...
متن کاملA pointfree version of remainder preservation
Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambd...
متن کاملProjective maximal submodules of extending regular modules
We show that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. Hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. As aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. This generalizes and simplifies a result of Dung and Smith. As another consequen...
متن کامل