Completion of a ring

Summary

The completion of a commutative ring $R$ with respect to an ideal $I$ is the ring $\hat R = \varprojlim R/I^n$. Its elements are compatible sequences of residues modulo the powers of $I$. It is the completion of $R$ for the $I$-adic topology, in the same way as the real numbers complete the rationals. The $p$-adic integers and the formal power series rings arise in this way. Completion is a basic tool to study a local ring near its closed point.