TITLE: Can a local ring have more than one prime ideal?
QUESTION [12 upvotes]: A local ring is defined as a ring which has a unique maximal ideal. This unique maximal ideal consists of only non-units and contains all the non-units of the ring $R$. So examples of local rings include any field, or rings localised at prime ideals and so on. 
My question is: can we have a local ring which has more than one prime ideal? I can't seem to think of any examples where this is the case.
(In response  to the comments below, I have realised that when $R$ is a domain, then $(0)$ is also a prime ideal, but are there any examples where $R$ isn't a domain?)

REPLY [8 votes]: For an explicit example, consider $k[[x,y]]$, formal power series over a field $k$, in two indeterminates. The unique maximal ideal is the set of series with no constant term; among the nonzero prime ideals are $(x)$ and $(y)$.