Let $R$ be a finite dimensional algebra over a field $K$.
Suppose there is a two-sided ideal $I$ of $R$ such that
(a) $R=K\oplus I$;
(b) $I$ is nilpotent.
Question
If $P$ is a left $R$-module, how does one define $\operatorname{Tor}_{i}^{R}(P,K)?$
I saw this done in a book (J-P Serre's Galois Cohomology) without comment. So, I suppose he is viewing $P$ as a right $R$-module somehow. (In the book he also supposes it is a finitely generated left $R$-module, but I think this is not used at this point).
One idea is to define the right $R$-multiplication by \begin{equation} p\cdot r=(r+I)p, \end{equation} that is, we quotient $r$ to its 'field representative' and view it inside $R$.
Looks like the properties hold and it is compatible with the left $R$-module structure (in the sense that it becomes a $R-R$ bimodule).
Is this it? Or maybe its a typo and it should be a right $R$-module?
It's also weird because later on he writes $P/IP$, suggesting that indeed $P$ is a left $R$-module. How is this done?
I thank in advance for the help.