The ring of real-valued functions which are continuous on a dense cozero set

Let $T''(X)$ and $T'(X)$ denote the collections of all real-valued functions on $X$ which are continuous on a dense cozero set and on an open dense subset of $X$ respectively. $T''(X)$ contains $C(X)$ and forms a subring of $T'(X)$ under pointwise addition and multiplication. We inquire when $T''(X)=C(X)$ and when $T''(X)=T'(X)$. We also ponder over the question when is $T''(X)$ isomorphic to $C(Y)$ for some topological space $Y$. We investigate some algebraic properties of the ring, $T''(X)$ for a Tychonoff space $X$. We provide several characterisations of $T''(X)$ as a Von-Neumann regular ring. We define nowhere almost $P$-spaces using the ring $T''(X)$ and characterise it as a Tychonoff space which has no non-isolated almost $P$-points. We show that a Tychonoff space with countable pseudocharacter is a nowhere almost $P$-space and highlight that this condition is not superflous using the closed ordinal space.