Levitzky's Theorem
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically ring theory and the theory of
nil ideal In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if all of its elements is nilpotent, i.e for each a \in I exists natural number ''n'' for which a^n = 0. If all elements of a ring ...
s, Levitzky's theorem, named after
Jacob Levitzki Jakob Levitzki, also known as Yaakov Levitsky (; 17 August 1904 – 25 February 1956), was an Israeli mathematician. Biography Levitzki was born in 1904 in the Ukrainian city, Kherson, then part of the Russian Empire. In 1912 he emigrated to the ...
, states that in a right
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
, every nil one-sided
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
is necessarily
nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...
. Levitzky's theorem is one of the many results suggesting the veracity of the
Köthe conjecture In mathematics, the Köthe conjecture is a problem in ring theory, open . It is formulated in various ways. Suppose that ''R'' is a ring. One way to state the conjecture is that if ''R'' has no nil ideal, other than , then it has no nil one-sided ...
, 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 Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
was given in .


Proof

This is Utumi's argument as it appears in ;Lemma Assume that ''R'' satisfies the
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
on annihilators of the form \ where ''a'' is in ''R''. Then # Any nil one-sided ideal is contained in the lower nil radical Nil*(''R''); # Every nonzero nil right ideal contains a nonzero nilpotent right ideal. # Every nonzero nil left ideal contains a nonzero nilpotent left ideal. ;Levitzki's Theorem Let ''R'' be a right Noetherian ring. Then every nil one-sided ideal of ''R'' is nilpotent. In this case, the upper and lower nilradicals are equal, and moreover this ideal is the largest nilpotent ideal among nilpotent right ideals and among nilpotent left ideals. ''Proof'': In view of the previous lemma, it is sufficient to show that the lower nilradical of ''R'' is nilpotent. Because ''R'' is right Noetherian, a maximal nilpotent ideal ''N'' exists. By maximality of ''N'', the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
''R''/''N'' has no nonzero nilpotent ideals, so ''R''/''N'' is a
semiprime ring In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as red ...
. As a result, ''N'' contains the lower nilradical of ''R''. Since the lower nilradical contains all nilpotent ideals, it also contains ''N'', and so ''N'' is equal to the lower nilradical.
Q.E.D. Q.E.D. or QED is an initialism of the List of Latin phrases (full), Latin phrase , meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of Mathematical proof ...


See also

*
Nilpotent ideal In mathematics, more specifically ring theory, an ideal ''I'' of a ring ''R'' is said to be a nilpotent ideal if there exists a natural number ''k'' such that ''I'k'' = 0. By ''I'k'', it is meant the additive subgroup generated by the set o ...
*
Köthe conjecture In mathematics, the Köthe conjecture is a problem in ring theory, open . It is formulated in various ways. Suppose that ''R'' is a ring. One way to state the conjecture is that if ''R'' has no nil ideal, other than , then it has no nil one-sided ...
*
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...


Notes


References

* * * * * * {{DEFAULTSORT:Levitzky's Theorem Theorems in ring theory