TITLE: Is there a simple way to state continuity for $I$-adic topology?
QUESTION [6 upvotes]: Let $R$ be a commutative ring with the $I$-adic topology defined by an ideal $I$, and let $S$ be a commutative ring with the $J$-adic topology for an ideal $J$. How would you translate saying that a homomorphism $f:R\to S$ is continuous? I am guessing it might be enough to say that for all $n>0$, there is an $m>0$ such that $f^{-1}(J^n)$ contains $I^m$. Is this right?

REPLY [5 votes]: A homomorphism is continuous iff it is continuous at $0$, and your condition is precisely the usual way to state the fact that a function is continuous at a point in terms of bases of neighborhoods at that point and at its image.