Non - Nilpotent Elements

Question

I am studying the theorem that " The nil radical of a ring R is the intersection of the prime ideals of R".

I understand that the set of nilpotent elements of R form an ideal.

What can we say about the set of non-nilpotent elements of R? It certainly contains 1, since 1 is not nilpotent, but is it closed under multiplication?

May you help me, please? Thank you in advance.

Answer 1

In $\mathbb Z_6$ we have $2$ and $3$.