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Nilradical equals intersection of prime ideals

In a commutative ring, the nilradical is the intersection of all prime ideals.

Nilradical theorem. Let RR be a commutative ring. Then

Nil(R)=⋂p∈Spec(R)p,\mathrm{Nil}(R)=\bigcap_{\mathfrak p\in \mathrm{Spec}(R)} \mathfrak p,

Thus an element is nilpotent if and only if it lies in every prime ideal. Equivalently, Nil(R)=(0)\mathrm{Nil}(R)=\sqrt{(0)}.

Remarks

The is the ideal of all . Since it is the of (0)(0), the quotient R/Nil(R)R/\mathrm{Nil}(R) is .