Nilpotent Element of a Ring

**meggnog** · July 22nd 2010, 04:51 PM

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.

**Jose27** · July 23rd 2010, 01:15 PM

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.

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