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In this article we define the $m$-topology on some rings of quotients of $C\left(X\right)$. Using this, we equip the classical ring of quotients $q\left(X\right)$ of $C\left(X\right)$ with the $m$-topology and we show that $C\left(X\right)$ with the $r$-topology is in fact a subspace of $q\left(X\right)$ with the $m$-topology. Characterization of the components of rings of quotients of $C\left(X\right)$ is given and using this, it turns out that $q\left(X\right)$ with the $m$-topology is connected if and only if $X$ is a pseudocompact almost $P$-space, if and only if $C\left(X\right)$ with $r$-topology is connected. We also observe that...