On \(z\)-ideals of pointfree function rings (Q464555)

An ideal \(I\) of a commutative ring \(A\) with identity is called a \(z\)-ideal, if whenever, two elements of \(A\) are in the same set of maximal ideals and \(I\) contains one of the elements, then it also contains the other. Let \(L\) be a completely regular frame and \(\mathcal{R}L\) denotes the ring of continuous real valued functions on \(L\). A frame is called a Yosida frame, if each of its compact elements is a meet of maximal elements. Some equivalent conditions for an ideal \(Q\) to be a \(z\)-ideal are given. Also a necessary and sufficient condition is given to show that which ideals of \(\mathcal{R}L\) are intersections of maximal ideals. It is shown that the lattice \(\mathrm{Zid}(L)\), of \(z\)-ideals of \(\mathcal{R}L\) partially ordered under set inclusion, is a normal coherent Yosida frame. It is also proved that (1) \(\mathrm{Zid}(\mathcal{R}L)\) is coherently normal and (2) \(\mathrm{Zid}(\mathcal{R}L)\) is regular iff \(L\) is a \(P\)-frame. Some results on commutative squares associated with \(z\)-ideals are proved. The following implications are proved. Let \(h : L \to M\) be a morphism in \textbf{CRegFrm}. Consider the following statements. {\parindent=6mm \begin{itemize}\item[1.] \(\mathrm{Zid}(h)\) is flat. \item[2.] \(\mathrm{Zid}(h)_{\ast}\) is a frame homomorphism. \item[3.] \(h\) is coz-flat. \end{itemize}}