A short note on the Marot property in rings of continuous functions

Let $X=Y\cup \left\{\omega\right\}$ where $\omega \notin Y$, topologized by equipping $Y$ with the discrete topology, and by letting deleted neighborhoods of $\omega$ consist of complements of closed discrete subsets of $Y$ in its Riemann surface topology. Assume that $I$ is an ideal of $C^{*}(X)$ where $C^{*}(X)$ is the ring of all bounded real-valued continuous functions on $X$. A result of Adler and Williams showed that $I$ contains a regular element if and only if a set of regular elements generates $I$. In this note, we obtain some conditions on $X$ for which the rings of continuous functions on $X$ are Marot. Moreover, this paper gives a sufficient condition for a quasi-B\'ezout ring to be additively regular.