algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

, the Amitsur–Levitzki theorem states that the algebra of ''n'' × ''n''

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

over a commutative ring satisfies a certain identity of

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

2''n''. It was proved by . In particular matrix rings are

polynomial identity ring In ring theory, a branch of mathematics, a ring ''R'' is a polynomial identity ring if there is, for some ''N'' > 0, an element ''P'' ≠ 0 of the free algebra, Z, over the ring of integers in ''N'' variables ''X''1, ''X''2, ..., ''X'N'' such th ...

s such that the smallest identity they satisfy has degree exactly 2''n''.

Statement

The standard polynomial of degree ''n'' is :

S_n(x_1,\dots,x_n) = \sum_\text(\sigma)x_ \cdots x_

in non-commuting variables ''x''₁, ..., ''x''_''n'', where the sum is taken over all ''n''! elements of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

''S''_''n''. The Amitsur–Levitzki theorem states that for ''n'' × ''n'' matrices ''A''₁, ..., ''A''_2''n'' whose entries are taken from a commutative ring then :

S_(A_1,\dots,A_) = 0.

Proofs

gave the first proof. deduced the Amitsur–Levitzki theorem from the Koszul–Samelson theorem about primitive cohomology of Lie algebras. and gave a simple

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

proof as follows. By linearity it is enough to prove the theorem when each matrix has only one nonzero entry, which is 1. In this case each matrix can be encoded as a directed edge of a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

with ''n'' vertices. So all matrices together give a graph on ''n'' vertices with 2''n'' directed edges. The identity holds provided that for any two vertices ''A'' and ''B'' of the graph, the number of odd

Eulerian path In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends ...

s from ''A'' to ''B'' is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges taken in order give an odd or even permutation of the 2''n'' edges.) Swan showed that this was the case provided the number of edges in the graph is at least 2''n'', thus proving the Amitsur–Levitzki theorem. gave a proof related to the

Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...

. gave a short proof using the

exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...

of a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

2''n''. gave another proof, showing that the Amitsur–Levitzki theorem is the Cayley–Hamilton identity for the generic Grassman matrix.

References

* * * * * * * * * * {{DEFAULTSORT:Amitsur-Levitzki theorem Linear algebra Theorems in algebra Matrix theory