Nilradical
Encyclopedia
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, more care is needed resulting in several related radicals.
of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent, and the product of any element with a nilpotent element is nilpotent. It can also be characterized as the intersection of all the prime ideal
s of the ring. If the ring is artinian
, the nilradical is its maximal nilpotent ideal.
–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.
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, the nilradical of a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
is the ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
consisting of the nilpotent elements of the ring. In the non-commutative ring case, more care is needed resulting in several related radicals.
Commutative rings
The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radicalRadical of an ideal
In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent, and the product of any element with a nilpotent element is nilpotent. It can also be characterized as the intersection of all the prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s of the ring. If the ring is artinian
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
, the nilradical is its maximal nilpotent ideal.
Noncommutative rings
For noncommutative rings, there are several analogues of the nilradical. The lower nilradical (or BaerReinhold Baer
Reinhold Baer was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings....
–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.