Nilpotent Element of a Ring

An element a of a commutative ring R is called nilpotent if for some n N.

Find the nilpotent elements in Z6, Z12, Z8, and Z36

Then show that the collection N of all nilpotent elements in R is an ideal.

For the first one, note that we have a non-zero nilpotent in iff the prime factorization of n exhibits at least one prime to a power distinct from 1. Can you then guess what elements are these nilpotents?

For the second use the binomial theorem.