# Nilpotent Element of a Ring

• Jul 22nd 2010, 04:51 PM
meggnog
Nilpotent Element of a Ring
An element a of a commutative ring R is called nilpotent if $a^n=0$ for some n $\epsilon$ 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.
• Jul 23rd 2010, 01:15 PM
Jose27
For the first one, note that we have a non-zero nilpotent in $\mathbb{Z} _n$ 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.