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!