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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<sub>1</sub>, X<sub>2</sub>, ..., X<sub>N</sub> such that

for all N-tuples r<sub>1</sub>, r<sub>2</sub>, ..., r<sub>N</sub> taken from R.

Strictly the X<sub>i</sub> 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.

Examples

:
  • 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<sub>N</sub>, 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<sub>i</sub> 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.
:
The m × m matrix ring over any commutative ring satisfies a standard identity: the Amitsur–Levitzki theorem states that it satisfies s<sub>2m</sub>. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than&nbsp;2m.
:e<sub>i</sub>&thinsp;e<sub>j</sub> = −e<sub>j</sub>&thinsp;e<sub>i</sub>.
This ring does not satisfy s<sub>N</sub> for any N and therefore can not be embedded in any matrix ring. In fact s<sub>N</sub>(e<sub>1</sub>,e<sub>2</sub>,...,e<sub>N</sub>) =&nbsp;N!e<sub>1</sub>e<sub>2</sub>...e<sub>N</sub>&nbsp;≠&nbsp;0. On the other hand it is a PI-ring since it satisfies [[x,&nbsp;y],&nbsp;z]&nbsp;:=&nbsp;xyz&nbsp;−&nbsp;yxz&nbsp;−&nbsp;zxy&nbsp;+&nbsp;zyx&nbsp;=&nbsp;0. It is enough to check this for monomials in the e<sub>i</sub>'s. Now, a monomial of even degree commutes with every element. Therefore if either x or y is a monomial of even degree [x,&nbsp;y] :=&nbsp;xy&nbsp;−&nbsp;yx&nbsp;=&nbsp;0. If both are of odd degree then [x,&nbsp;y]&nbsp;=&nbsp;xy&nbsp;−&nbsp;yx&nbsp;=&nbsp;2xy has even degree and therefore commutes with z, i.e. [[x,&nbsp;y],&nbsp;z]&nbsp;=&nbsp;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<sub>N</sub> for N&nbsp;>&nbsp;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 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 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&nbsp;:=&nbsp;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

References

Further reading

External links