 Adic topology

For a two-sided ideal $I$ of a ring $R$, the powers $I^n$ form a neighborhood base at zero for the adic topology. For an $R$-module $M$, use $I^{n}M$ on the left or $M{I}^n$ on the right. Convergence means that for every $n$ the differences eventually belong to the corresponding submodule. The formal power series ring $k[[t]]$ and formal power series module $M[[t]]$ are complete for $I=(t)$; convergence fixes each finite list of coefficients eventually. This is formal convergence, without an analytic condition on the size of coefficients.