Nilpotents, Ideals and Nilradicals
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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
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Solution Summary
Nilpotents, Ideals and Nilradicals are investigated.
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Proof:
Since each ai is a nilpotent element, then we can find some positive integer ri for each ai, such that ai^ri=0. Now let n=lcm{r1,r2,...,rm} be the least ...
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