Take the tour ×  Let $R$ be a Noetherian domain, and let
$\mathfrak{p}$ be a prime ideal; consider the completion $\hat
R_{\mathfrak{p}}$ of $R$ at $\mathfrak{p}$ (the inverse limit of
the system of quotients $R/\mathfrak{p}^n$). If $R$ is a PID, it
is easy to see that $\hat R_{\mathfrak{p}}$ is a domain.
Someone asked in sci.math if $\hat R_{\mathfrak{p}}$ would
always be a domain. I thought it would, but looking at
Eisenbud's "Commutative Algebra", I found a reference to a
theorem of Larfeldt and Lech that says that if $A$ is any
finite-dimensional algebra over a field $k$, then there is a
Noetherian local integral domain $R$ with maximal ideal
$\mathfrak{m}$ such that $\hat{R_{\mathfrak{M}}}\cong
A[[x_1,\ldots,x_n]]$ for some $n$; and so this completion will
not be a domain if $A$ is not a domain. I would like to know an
example directly, if possible.  Does someone know an easy
example of a noetherian domain $R$ and a prime ideal
$\mathfrak{p}$ such that $\hat R_{\mathfrak{p}}$ is not a
domain? Thanks in