Let
be the set of all nilpotent elements of a commutative ring
, and show that
for each prime ideal
of
.
Proof: Let
and
. Then there is a positive integer
with
. Assume towards a contradiction that
. Since
and
, then by definition of prime ideals we have
. However, the same can be said for any integer
, that is, if
then
and therefore
. By induction, since
, then
, which contradicts our assumption that
. So
whenever
, that is,
.
The reasoning is sound, but I'm not quite sure how to best articulate it. I don't think "by induction" is completely appropriate, here. Also, I like to avoid by-contradiction proofs if possible. Does anyone have an idea for how I can more elegantly and accurately explain what's going on in this exercise?
Thanks!