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Let $A$ be a commutative ring with identity and $I$ an ideal of $A$. $A$ is said to be $I$-$clean$ if for every element $a\in A$ there is an idempotent $e=e^2\in A$ such that $a-e$ is a unit and $ae$ belongs to $I$. A filter of ideals, say $ℱ$, of $A$ is if for each $I\in ℱ$ there is a finitely generated ideal $J\in ℱ$ such that $J\subseteq I$. We characterize $I$-clean rings for the ideals $0$, $n\left(A\right)$, $J\left(A\right)$, and $A$, in terms of the frame of multiplicative Noetherian filters of ideals of $A$, as well as in terms of more classical ring properties.

Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the $m$-topology on $C\left(X\right)$, denoted $C_m\left(X\right)$, and demonstrated that certain topological properties of $X$ could be characterized by certain topological properties of $C_m\left(X\right)$. For example, he showed that $X$ is pseudocompact if and only if $C_m\left(X\right)$ is a metrizable space; in this case the $m$-topology is precisely the topology of uniform convergence. What is interesting with regards to the $m$-topology is that it is possible, with...