Completion of a ring

Remarks

The map $R \to \hat R$, $r \mapsto (r + I^n)_n$, is a ring homomorphism with kernel $\bigcap_n I^n$. Equivalently, $\hat R$ is the ring of Cauchy sequences for the $I$-adic topology modulo the sequences that tend to $0$. For an $R$-module $M$ one defines $\hat M = \varprojlim_n M/I^n M$ in the same way. Example: for $R = k[x]$, with $k$ a field, and $I = (x)$, a compatible sequence is the list of truncations of a formal power series, and $\hat R \cong k[\![x]\!]$. Non-example: for $R = \Z$ and $I = 2\Z$, the sequence starting with $x_1 = 0 + 2\Z$ and $x_2 = 1 + 4\Z$ does not give an element of $\hat R$, since $\pi_{21}(x_2) = 1 + 2\Z \neq x_1$.