Take the 2-minute tour ×  Let $R$ be a non-Noetherian local
commutative ring with identity such that it is of Krull-
dimension zero. I am wondering if there are conditions which
will force the maximal ideal to be nilpotent.  share|improve
this question What do you mean by "the only prime ideal"? There
is no reason why there is only one prime ideal. –  Martin
Brandenburg Feb 13 '13 at 20:03 Martin is absolutely right: if
$R$ has only one prime ideal, it is of dimension zero! (And then
that prime ideal indeed consists of nilpotent elements but
needn't be nilpotent) –  Georges Elencwajg Feb 13 '13 at 20:23
Since your ring is local, did you perhaps mean to type "the only
$maximal$ ideal"? I think this would make your question
interesting. –  mbrown Feb 13 '13 at 21:30 I must apologize. I
was a bit confused by the definition of the dimension in a ring.
Yes, it should be 0-dimensional ring. So we have a 0-dimensional
local ring. Then the maximum ideal's elements are nilpotent. I
was wondering if there are conditions