Locally nilpotent
Encyclopedia
In the mathematical
field of commutative algebra
, an ideal
I in a commutative ring
A is locally nilpotent at a prime ideal
p if the ideal is nilpotent in a Zariski open neighborhood
of p. That is, I is nilpotent at p if Ip, the localization
of I at p, is a nilpotent ideal
in Ap.
In non-commutative algebra and group theory, an algebra or group is locally nilpotent if and only if every finitely generated subalgebra or subgroup is nilpotent. The subgroup generated by the normal locally nilpotent subgroups is called the Kurt Hirsch
-Plotkin radical and is the generalization of the Fitting subgroup
to groups without the ascending chain condition on normal subgroups. In non-commutative ring theory, a locally nilpotent ideal is called a nil ideal
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, an 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"....
I in 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....
A is locally nilpotent at a 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...
p if the ideal is nilpotent in a Zariski open neighborhood
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
of p. That is, I is nilpotent at p if Ip, the localization
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
of I at p, is a nilpotent ideal
Nilpotent ideal
In mathematics, more specifically ring theory, an ideal, I, of a ring is said to be a nilpotent ideal, if there exists a natural number k such that Ik = 0. By Ik, it is meant the additive subgroup generated by the set of all products of k elements in I...
in Ap.
In non-commutative algebra and group theory, an algebra or group is locally nilpotent if and only if every finitely generated subalgebra or subgroup is nilpotent. The subgroup generated by the normal locally nilpotent subgroups is called the Kurt Hirsch
Kurt Hirsch
Kurt August Hirsch was a German mathematician. He studied at the University of Berlin where he was taught by Bieberbach, von Mises, Schmidt, and Schur. Although most influenced by Schur, his doctoral dissertation was on the philosophy of mathematics. The thesis examines the 1920s dispute between...
-Plotkin radical and is the generalization of the Fitting subgroup
Fitting subgroup
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable...
to groups without the ascending chain condition on normal subgroups. In non-commutative ring theory, a locally nilpotent ideal is called a nil ideal
Nil ideal
In mathematics, more specifically ring theory, an ideal of a ring is said to be a nil ideal if each of its elements is nilpotent. The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil...
.