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Abstract Algebra

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Moderate

Nilradical of a Ring: Relation to Prime Ideals

ABSALG-0ML3ZL

Let $R$ be a commutative ring.

The nilradical $\text{nil}(R )$ of $R$ is equal to the intersection of all prime ideals of $R$.

Which of the following are equal to $\text{nil}(R )$?

Select ALL that apply.

A

The intersection of all prime ideals containing $0$.

B

The intersection of all prime ideals containing $1$.

C

The intersection of all ideals containing a prime ideal.

D

The intersection of all maximal ideals.

E

The ideal of $R$ consisting of all nilpotent elements of $R$