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.
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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.
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