Levitzky's theorem
Encyclopedia
In mathematics
, more specifically ring theory
and the theory of nil ideal
s, Levitzky's theorem, named after Jacob Levitzki
, states that in a right Noetherian ring
, every nil one-sided ideal is necessarily nilpotent
. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indeed provided a solution to one of Köthe's questions as described in . The result was originally submitted in 1939 as , and a particularly simple proof was given in .
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...
, more specifically ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
and the theory of 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...
s, Levitzky's theorem, named after Jacob Levitzki
Jacob Levitzki
Jacob Levitzki, also known as Yaakov Levitsky was an Israeli mathematician.-Biography:Levitzki was born in 1904 in Ukraine, then part of the Russian Empire, and emigrated to then Ottoman-ruled Palestine in 1912...
, states that in a right Noetherian ring
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
, every nil one-sided ideal is necessarily nilpotent
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...
. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indeed provided a solution to one of Köthe's questions as described in . The result was originally submitted in 1939 as , and a particularly simple proof was given in .