On the sum of \(z^\circ\)-ideals in two classes of subrings of \(C(X)\)
Let \(X\) be a Tychonoff space, \(C(X)\) be the ring of real-valued continuous functions on \(X\), \(I\) be an ideal of a subring \(A(X)\) of \(C(X)\) and \(C^*(X)\subseteq A(X)\). Then \(I\) is called a \(z^\circ\)-ideal if for every \(a \in I,~ P_a \subseteq I\), where \(P_a\) denotes the intersection of all minimal prime ideals of \(A(X)\) that contain \(a\). Generally, the goal of this paper is to consider sums of \(z^\circ\)-ideals in intermediate rings of \(C(X)\). Recall that \(X\) is called an \(F\)-space if every cozero-set in \(X\) is \(C^*\)-embedded. Algebraically, \(X\) is an \(F\)-space if and only if every finitely generated ideal of \(C(X)\) is principal. Recall that if every dense cozero-set is \(C^*\)-embedded in \(X\), then \(X\) is called a quasi \(F\)-space. In this paper it is proved that \(X\) is a quasi \(F\)-space if and only if the sum of any two \(z^\circ\)-ideals in \(A(X)\) (any intermediate ring) is a \(z^\circ\)-ideal or all of \(A(X)\) (the whole intermediate ring). Let \(X\) be an almost \(P\)-space. Then, \(X\) is an \(F\)-space if and only if for each ideal \(I\) in \(C(X)\), such that \(I+R\) generates the topology on \(X\), the sum of any two \(z^\circ\)-ideals in \(I+\mathbb{R}\) is a \(z^\circ\)-ideal in \(I+\mathbb{R}\) or all of \(I+\mathbb{R}\).
scientific article; zbMATH DE number 7377351
scientific article