Let X be a Tychonoff space and ${sum}(X)$ the set of all the subrings of C(X) that contain $C^*(X)$. For any A(X) in ${sum}(X)$ suppose $_{{upsilon}A}X$ is the largest subspace of ${eta}X$ containing X to which each function in A(X) can be extended continuously. Let us write A(X) ~ B(X) if and only if $_{{upsilon}A}X=_{{upsilon}B}X$, thereby defining an equivalence relation on ${sum}(X)$. We have shown that an A(X) in ${sum}(X)$ is isomorphic to C(Y ) for some space Y if and only if A(X) is the largest member of its equivalence class if and only if there exists a subspace T of ${eta}X$ with the property that A(X)={$f{in}C(X):f^*(p)$ is real for each $p$ in T}, $f^*$ being the unique continuous extension of $f$ in C(X) from ${eta}X$ to $mathbb{R}^*$, the one point compactification of $mathbb{R}$. As a consequence it follows that if X is a realcompact space in which every $C^*$-embedded subset is closed, then C(X) is never isomorphic to any A(X) in ${sum}(X)$ without being equal to it.