Letting R be a commutative ring, and letting I = {a in R | a^n =0 for some n in S} where S is the set of positive integers. Prove I is an ideal of R.
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Originally Posted by Coda202 Letting R be a commutative ring, and letting I = {a in R | a^n =0 for some n in S} where S is the set of positive integers. Prove I is an ideal of R. This is just saying that the nilradical is an ideal. note if $\displaystyle x^n=0$ then $\displaystyle \forall a \in R$ $\displaystyle (ax)^n=0$ Also if $\displaystyle x^n=y^n=0$ then $\displaystyle (x+y)^{m+n-1}= \sum_{k=0}^{m+n-1}{n \choose k} x^{m+n-1-k}y^k=0.$
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