In
ring theory, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 that
:
for all ''N''-
tuples ''r''
1, ''r''
2, ..., ''r''
''N'' taken from ''R''.
Strictly the ''X''
''i'' here are "non-commuting indeterminates", and so "polynomial identity" is a slight
abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring ''S'' may be used, and gives the concept of PI-algebra.
If the
degree of the polynomial ''P'' is defined in the usual way, the polynomial ''P'' is called monic if at least one of its terms of highest degree has coefficient equal to 1.
Every
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
is a PI-ring, satisfying the polynomial identity ''XY'' − ''YX'' = 0. Therefore, PI-rings are usually taken as ''close generalizations of commutative rings''. If the ring has
characteristic ''p'' different from zero then it satisfies the polynomial identity ''pX'' = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.
[J.C. McConnell, J.C. Robson, ''Noncommutative Noetherian Rings, ]Graduate Studies in Mathematics Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats.
List of books
*1 ''The General To ...
, Vol 30''
Examples
* For example, if ''R'' is a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
it is a PI-ring: this is true with
::
*The ring of 2 × 2
matrices over a commutative ring satisfies the Hall identity
::
:This identity was used by , but was found earlier by .
* A major role is played in the theory by the standard identity ''s''
''N'', of length ''N'', which generalises the example given for commutative rings (''N'' = 2). It derives from the
Leibniz formula for determinants
::
:by replacing each product in the summand by the product of the ''X''
''i'' in the order given by the
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
σ. In other words each of the ''N'' ! orders is summed, and the coefficient is 1 or −1 according to the
signature.
::
:The ''m'' × ''m''
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
over any commutative ring satisfies a standard identity: the
Amitsur–Levitzki theorem states that it satisfies ''s''
2''m''. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2''m''.
* Given a
field ''k'' of characteristic zero, take ''R'' to be the
exterior algebra over a
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
-
dimensional vector space with
basis ''e''
1, ''e''
2, ''e''
3, ... Then ''R'' is generated by the elements of this basis and
::''e''
''i'' ''e''
''j'' = − ''e''
''j'' ''e''
''i''.
:This ring does not satisfy ''s''
''N'' for any ''N'' and therefore can not be
embedded in any matrix ring. In fact ''s''
''N''(''e''
1,''e''
2,...,''e''
''N'') = ''N'' ! ''e''
1''e''
2...''e''
''N'' ≠ 0. On the other hand it is a PI-ring since it satisfies
''x'', ''y'' ''z''] := ''xyz'' − ''yxz'' − ''zxy'' + ''zyx'' = 0. It is enough to check this for monomials in the ''e''
''i'''s. Now, a monomial of
parity (mathematics), even degree commutes with every element. Therefore if either ''x'' or ''y'' is a monomial of even degree
'x'', ''y'':= ''xy'' − ''yx'' = 0. If both are of
odd degree then
'x'', ''y''nbsp;= ''xy'' − ''yx'' = 2''xy'' has even degree and therefore commutes with ''z'', i.e.
''x'', ''y'' ''z''] = 0.
Properties
* Any
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...
or
homomorphic
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of a PI-ring is a PI-ring.
* A finite
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of PI-rings is a PI-ring.
* A direct product of PI-rings, satisfying the same identity, is a PI-ring.
* It can always be assumed that the identity that the PI-ring satisfies is
multilinear.
* If a ring is
finitely generated by ''n'' elements as a
module over its
center then it satisfies every alternating multilinear polynomial of degree larger than ''n''. In particular it satisfies ''s''
''N'' for ''N'' > ''n'' and therefore it is a PI-ring.
* If ''R'' and ''S'' are PI-rings then their
tensor product over the integers,
, is also a PI-ring.
* If ''R'' is a PI-ring, then so is the ring of ''n'' × ''n'' matrices with coefficients in ''R''.
PI-rings as generalizations of commutative rings
Among non-commutative rings, PI-rings satisfy the
Köthe conjecture.
Affine PI-algebras over a
field satisfy the
Kurosh conjecture, the
Nullstellensatz and the
catenary property for
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s.
If ''R'' is a PI-ring and ''K'' is a subring of its center such that ''R'' is
integral over In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' i ...
''K'' then the
going up and going down properties for prime ideals of ''R'' and ''K'' are satisfied. Also the ''lying over'' property (If ''p'' is a prime ideal of ''K'' then there is a prime ideal ''P'' of ''R'' such that
is minimal over
) and the ''incomparability'' property (If ''P'' and ''Q'' are prime ideals of ''R'' and
then
) are satisfied.
The set of identities a PI-ring satisfies
If ''F'' := Z is the free algebra in ''N'' variables and ''R'' is a PI-ring satisfying the polynomial ''P'' in ''N'' variables, then ''P'' is in the
kernel of any homomorphism
:
: ''F''
''R''.
An
ideal ''I'' of ''F'' is called T-ideal if
for every
endomorphism ''f'' of ''F''.
Given a PI-ring, ''R'', the set of all polynomial identities it satisfies is an
ideal but even more it is a T-ideal. Conversely, if ''I'' is a T-ideal of ''F'' then ''F''/''I'' is a PI-ring satisfying all identities in ''I''. It is assumed that ''I'' contains monic polynomials when PI-rings are required to satisfy monic polynomial identities.
See also
*
Posner's theorem
*
Central polynomial In Abstract algebra, algebra, a central polynomial for ''n''-by-''n'' Matrix (mathematics), matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at ''n''-by-''n'' matrices. That ...
References
*
*
Polynomial identities in ring theory Louis Halle Rowen, Academic Press, 1980,
Polynomial identity rings Vesselin S. Drensky, Edward Formanek, Birkhäuser, 2004,
Polynomial identities and asymptotic methods A. Giambruno, Mikhail Zaicev, AMS Bookstore, 2005,
Computational aspects of polynomial identities Alexei Kanel-Belov, Louis Halle Rowen, A K Peters Ltd., 2005,
Further reading
*
*
External links
*
*
*
{{DEFAULTSORT:Polynomial Identity Ring
Ring theory