\(C\) and \(C^*\) among intermediate rings (Q2862923)

For a completely regular space \(X\), \(A(X)\) is assumed to be a ring of real valued continuous functions between \(C^{*}(X)\) and \(C(X)\). This class of rings has been studied by many authors. By \textit{H. L. Byun} and \textit{S. Watson} [Topology Appl. 40, No. 1, 45--62 (1991; Zbl 0732.54016)], many properties of \(C(X)\) and \(C^{*}(X)\) are true for \(A(X)\). The correspondence between ideals of \(C^{*}(X)\) and \(z\)-filters was generalized to intermediate rings, see [\textit{L. Redlin} and \textit{S. Watson}, Proc. Am. Math. Soc. 100, 763--766 (1987; Zbl 0622.54011); Fundam. Math. 152, No. 2, 151--163 (1997; Zbl 0877.54015)] and [Byun and Watson, loc. cit.]. The correspondence between ideals of \(C(X)\) and \(z\)-filters was generalized to all intermediate rings by \textit{P. Panman, J. Sack} and \textit{S. Watson} [Correspondences between ideals and \(z\)-filters for rings of continuous functions between \(C\) and \(C^*\), Commentat. Math. 52, 11--20 (2012)].NEWLINENEWLINEIn this paper, the authors first give a description of the old results. Afterwards they introduce \(\mathcal{Z}_{A}\) (resp., \(\mathcal{E}_{A}\) )-ideals and \(\mathcal{Z}_{A}\) (resp., \(\mathcal{E}_{A}\) )-filters. Then they investigate the relationship between maximal ideals of \(A(X)\) and \(\mathcal{Z}_{A}\) (resp., \(\mathcal{E}_{A}\) )-ultrafilters on \(X\). The definition of \(\mathcal{E}_{A}\)-ideal is new but \(\mathcal{Z}_{A}\) is the \(\mathcal{B}\)-ideal which was introduced in [Byun and Watson, loc. cit.]. The paper also obtains some interesting questions. Finally, we know that the sum of two \(z\)-ideals in \(C(X)\) is a \(z\)-ideal. This motivates the following questions in this paper. 1. Is the sum of two \(\mathcal{Z}_{A}\)-ideals in \(A(X)\) a \(\mathcal{Z}_{A}\)-ideal? 2. Is the sum of two \(\mathcal{E}_{A}\)-ideals in \(A(X)\) a \(\mathcal{E}_{A}\)-ideal?