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Prove that if R is commutative ring and N=(a1,a2,

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Prove that if R is commutative ring and N=(a1,a2,..am) where each ai is a nilpotent element, then N is a nilpotent ideal, i.e N^n=0 for some positive integer n. Deduce that if the nilradical of R is finitely generated then it is a nilpotent ideal.

P.S. the set of nilpotent elements form an ideal which is called nilradical of R for a commutative ring R