Assume $\mathcal{A}$ is a small cocomplete abelian $\otimes$-category (see here for a definition). Is there a cocontinuous, full, faithful, exact $\otimes$-functor $\mathcal{A} \to \text{Mod}(R)$ for some ring $R$?

See here for a related question. The reason why I'm asking is that I want to prove that a certain (perfectly natural!) construction works in such a category $\mathcal{A}$ and it seems to me that the axioms of $\mathcal{A}$ are not enough to check that a certain sequence is exact. In module categories there are no problems.

ringyou meancommutative ring? – Mariano Suárez-Alvarez♦ Nov 25 '10 at 17:24