Nilradical of a ring
In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring.
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 of 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 (by commutativity) is nilpotent. 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.)
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 is called a Jacobson ring if the nilradical of R/P coincides with the Jacobson radical of R/P for every prime ideal 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.
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