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Concept
Nilradical of a ring
Summary
In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements:
It is thus the radical of the zero ideal. If the nilradical is the zero ideal, the ring is called a reduced ring. The nilradical of a commutative ring is the intersection of all prime ideals.
In the non-commutative ring case the same definition does not always work. This has resulted in several radicals generalizing the commutative case in distinct ways; see the article Radical of a ring for more on this.
The nilradical of a Lie algebra is similarly defined for Lie algebras.
The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent (by the binomial formula), and the product of any element with a nilpotent element is nilpotent (by commutativity). It can also be characterized as the intersection of all the prime ideals of the ring (in fact, it is the intersection of all minimal prime ideals).
Let be a commutative ring. Then
Let and be a prime ideal, then for some . Thus
since is an ideal, which implies or . In the second case, suppose for some , then thus or and, by induction on , we conclude , in particular . Therefore is contained in any prime ideal and .
Conversely, we suppose and consider the set which is non-empty, indeed . is partially ordered by and any chain has an upper bound given by , indeed: is an ideal and if for some then for some , which is impossible since ; thus any chain in has an upper bound and we can apply Zorn's lemma: there exists a maximal element . We need to prove that is a prime ideal: let , then since is maximal in , which is to say, there exist such that , but then , which is absurd. Therefore if , is not contained in some prime ideal or equivalently and finally .
A ring is called reduced if it has no nonzero nilpotent. Thus, a ring is reduced if and only if its nilradical is zero.
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