A ring of cohomological operators on Ext and Tor

Let $f\text{\hspace{0.17em}⁣}:R\to B$ be a surjective homomorphism of rings with kernel $I$. Gulliksen (when $I$ is generated by a regular sequence) and later Mehta (in general) showed that for any $B$-modules $M$ and $N$, ${\mathrm{E}\mathrm{x}\mathrm{t}}_B^{\ast }(M,N)$ has a structure of graded ${\mathrm{S}}_B^{\ast }\left(\widehat{I\mathrm{/}{I}^2}\right)$-module, where $\widehat{}$ denotes dual and $\mathrm{S}$ denotes symmetric algebra. This construction is extended to the case where $f$ is not necessarily surjective in a way that allows one to regard these operators from a more natural perspective.