Nilradical of a ring
In
It is thus the
In the
The nilradical of a Lie algebra is similarly defined for Lie algebras.
Commutative rings
The nilradical of a commutative ring is the set of all
Proposition[1] — Let be a commutative ring. Then the nilradical of equals the intersection of all prime ideals of
Firstly, the nilradical is contained in every prime ideal. Indeed, if one has for some positive integer Since every ideal contains 0 and every prime ideal that contains a product, here contains one of its factors, one deduces that every prime ideal contains
Conversely, let we have to prove that there is a prime ideal that does not contains Consider the set of all ideals that do not contain any power of One has by definition of the nilradical. For every chain of ideals in the union is an ideal that belongs to since otherwise it would contain a power of that must belong to some contradicting the definition of
So, is a
A ring is called reduced if it has no nonzero nilpotent. Thus, a ring is reduced if and only if its nilradical is zero. If R is an arbitrary commutative ring, then the quotient of it by the nilradical is a reduced ring and is denoted by .
Since every maximal ideal is a prime ideal, the Jacobson radical — which is the intersection of maximal ideals — must contain the nilradical. A ring R is called a Jacobson ring if the nilradical and Jacobson radical of R/P coincide for all prime ideals P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.
Noncommutative rings
For noncommutative rings, there are several analogues of the nilradical. The lower nilradical (or Baer–McCoy radical, or prime radical) is the analogue of the radical of the zero ideal and is defined as the intersection of the prime ideals of the ring. The analogue of the set of all nilpotent elements is the upper nilradical and is defined as the ideal generated by all nil ideals of the ring, which is itself a nil ideal. The set of all nilpotent elements itself need not be an ideal (or even a subgroup), so the upper nilradical can be much smaller than this set. The Levitzki radical is in between and is defined as the largest locally nilpotent ideal. As in the commutative case, when the ring is Artinian, the Levitzki radical is nilpotent and so is the unique largest nilpotent ideal. Indeed, if the ring is merely Noetherian, then the lower, upper, and Levitzki radical are nilpotent and coincide, allowing the nilradical of any Noetherian ring to be defined as the unique largest (left, right, or two-sided) nilpotent ideal of the ring.
References
- ISBN 0-201-40751-5., p.5
- ISBN 0-387-94268-8.
- MR 1838439