Completion of a Polynomial Ring

Question

I want to show that the ring of formal power series $R[[X]]$ is the completion of $R[X]$ with respect to the metric $d\colon R[[X]] \times R[[X]] \rightarrow \mathbb{R}, d(f,g) = 2^{\min \{i|f_i\ne g_i\}}$. Since, I can not find a strict definition for what completion even means, I have a problem to start with. Is the definition the same as when working with vector spaces in functional analysis?

Answer 1

In Algebra the completion of a ring $R$ with respect to an ideal $I$ (a.k.a. the $I$-adic completion of $R$) is the limit $\hat R$, in the algebraic sense, of the quotient rings $R/I^n$ with respect to the canonical maps $$ \pi_n:R/I^{n+1}\longrightarrow R/I^n, $$ i.e., an element $r\in\hat R$ is but a sequence $r_n\in R/I^n$ such that $\pi_n(r_{n+1})=r_n$. It is a straightforward exercise to show that $\hat R$ is a ring and that there's a canonical map of rings $\Phi:R\rightarrow \hat R$.

When $R$ is a topological ring the quotient topology in each quotient $R/I_n$ defines a topology in $\hat R$. If $R$ is Hausdorff and $\bigcap_nI^n=(0)$ then

• $\hat R$ is Hausdorff,
• $\Phi$ is an embedding.

Moreover, if $R$ is metric and $I^n$ can be realized as the ball centered in $0$ of radius $1/n$ then $\hat R$ is the completion of $R$ in the "usual" sense, namely the set of equivalence classes of Cauchy sequences in $R$.

The ring of formal power series $K[[X]]$ is the $(X)$-adic completion of the ring of polynomials $K[X]$.

Another very important example is the ring of $p$-adic integers $\Bbb Z_p$ which is the $p$-adic completion of the ring integers $\Bbb Z$ with respect to the maximal ideal generated by the prime $p$. The field $\Bbb Q_p$ of $p$-adic numbers is the field of fractions of $\Bbb Z_p$ and can be obtianed directly as the completion (in the metric sense) of $\Bbb Q$ under the $p$-adic metric induced by the $p$-adic absolute value.