A topological ring is a ring equipped with a topology making addition, additive inversion and multiplication continuous functions. An adic topology is obtained from powers of an ideal.
For a two-sided ideal of a ring , the powers form a neighborhood base at zero for the adic topology. For an -module , use on the left or on the right. Convergence means that for every the differences eventually belong to the corresponding submodule. The formal power series ring and formal power series module are complete for ; convergence fixes each finite list of coefficients eventually. This is formal convergence, without an analytic condition on the size of coefficients.
The adic completion is the inverse limit of the quotient modules . An element is a compatible family of residue classes at every order. For and , this gives the formal power series module . The canonical map from need not be injective in general: its kernel is . It is injective for this polynomial example because a nonzero polynomial has finite degree. Successive changes of coordinates in formal rigidity from vanishing second Hochschild cohomology converge in this completion because the order- change is the identity modulo .
