Noun
nilradical (plural nilradicals)
(algebra) The set of nilpotent elements of an algebraic structure such as an ideal
Elements of the Jacobson radical and nilradical can be therefore seen as generalizations of 0. Equivalent characterizations The Jacobson radical of a ring has various internal and external characterizations. Source: Internet
In fact for any ring, the nilpotent elements in the center of the ring are also in the Jacobson radical.sfn So, for commutative rings, the nilradical is contained in the Jacobson radical. Source: Internet
These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring's Jacobson radical. Source: Internet
If N is the nilradical of commutative ring R, then the quotient ring R/N has no nilpotent elements. Source: Internet
The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring's minimal primes is zero. Source: Internet