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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.

The intersection of all prime ideals containing $0$.

The intersection of all prime ideals containing $1$.

The intersection of all ideals containing a prime ideal.

The intersection of all maximal ideals.

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