1. ## Nilpotents

Letting R be a commutative ring, and letting I = {a in R | a^n =0 for some n in S} where S is the set of positive integers.
Prove I is an ideal of R.

2. Originally Posted by Coda202
Letting R be a commutative ring, and letting I = {a in R | a^n =0 for some n in S} where S is the set of positive integers.
Prove I is an ideal of R.
This is just saying that the nilradical is an ideal.

note if $x^n=0$ then $\forall a \in R$ $(ax)^n=0$

Also if $x^n=y^n=0$ then

$(x+y)^{m+n-1}= \sum_{k=0}^{m+n-1}{n \choose k} x^{m+n-1-k}y^k=0.$