Let $R$ be a ring with a descending filtration by ideals
$$R = I^{(0)} \supset I^{(1)} \supset I^{(2)} \supset \dots$$
such that $I^{(j)} I^{(k)} \subset I^{(j+k)}$, with equality whenever $k$ is sufficiently large. Is the topology induced by this filtration necessarily the same as the $\mathfrak a$-adic topology for some ideal $\mathfrak a \subset R$?