Nilradical is an ideal
Commutativity is crucial to the proof, as we shall see.
Abelian group structure
It is clear that is nilpotent, and that if is nilpotent, so is . We thus only need to show closure under addition. Suppose and are nilpotent with . Then consider:
We can expand this by the binomial theorem. We get a sum of monomials. For each monomial, either the power of is at least or the power of is at least . Thus, each of the monomials in the expansion is zero, and so the above expression simplifies to 0.
Commutativity is essentially to rewrite expressions like as a power of , times a power of .
The ideal property
We need to show that if is nilpotent, and is any ring element, then is nilpotent. Since there exists a such that , we have . Here, we again use commutativity to move the s past the s.