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 that
:$P(r_1, r_2, \ldots, r_N) = 0$
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 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, Vol 30''

Examples

* For example, if ''R'' is a commutative ring it is a PI-ring: this is true with
::$P(X_1,X_2) = X_1X_2-X_2X_1 = 0~$

*The ring of 2 × 2 matrices over a commutative ring satisfies the Hall identity
::$(xy-yx)^2z=z(xy-yx)^2$
: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

::$\det(A) = \sum_ \sgn(\sigma) \prod_^N a_$

:by replacing each product in the summand by the product of the ''X''_''i'' in the order given by the permutation σ. In other words each of the ''N'' ! orders is summed, and the coefficient is 1 or −1 according to the signature.

::$s_N(X_1,\ldots,X_N) = \sum_ \sgn(\sigma) X_\dotsm X_=0~$

:The ''m'' × ''m'' matrix ring 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-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 or homomorphic image of a PI-ring is a PI-ring.
* A finite direct product 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, $R\otimes_\mathbbS$, 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 ideals.

If ''R'' is a PI-ring and ''K'' is a subring of its center such that ''R'' is integral over ''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 $p$ is minimal over $P\cap K$) and the ''incomparability'' property (If ''P'' and ''Q'' are prime ideals of ''R'' and $P\subset Q$ then $P\cap K\subset Q\cap K$) 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
:$\tau$: ''F'' $\rightarrow$ ''R''.

An ideal ''I'' of ''F'' is called T-ideal if $f(I)\subset I$ 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

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

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External links

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{{DEFAULTSORT:Polynomial Identity Ring
Ring theory