# Polynomial identity ring

In mathematics, in the subfield of ring theory, a ring *R* is a **polynomial identity ring** if there is, for some *N* > 0, an element *P* other than 0 of the free algebra, Z⟨*X*_{1}, *X*_{2}, ..., *X*_{N}⟩, 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* it happens that

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.[1]

## Examples

- For example, if
*R*is a commutative ring it is a PI-ring: this is true with

- The ring of 2 by 2 matrices over a commutative ring satisfies the
**Hall identity**

- 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 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*_{2m}. 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'*s. Now, a monomial of 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*] =*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, , 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 noncommutative 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* := Z⟨*X*_{1}, *X*_{2}, ..., *X*_{N}⟩ 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

- J.C. McConnell, J.C. Robson,
*Noncommutative Noetherian Rings, Graduate Studies in Mathematics, Vol 30*

- Latyshev, V.N. (2001) [1994], "PI-algebra", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Formanek, E. (2001) [1994], "Amitsur–Levitzki theorem", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Polynomial identities in ring theory, Louis Halle Rowen, Academic Press, 1980, ISBN 978-0-12-599850-5
- Polynomial identity rings, Vesselin S. Drensky, Edward Formanek, Birkhäuser, 2004, ISBN 978-3-7643-7126-5
- Polynomial identities and asymptotic methods, A. Giambruno, Mikhail Zaicev, AMS Bookstore, 2005, ISBN 978-0-8218-3829-7
- Computational aspects of polynomial identities, Alexei Kanel-Belov, Louis Halle Rowen, A K Peters Ltd., 2005, ISBN 978-1-56881-163-5

## Further reading

- Formanek, Edward (1991).
*The polynomial identities and invariants of*n*×*n*matrices*. Regional Conference Series in Mathematics.**78**. Providence, RI: American Mathematical Society. ISBN 0-8218-0730-7. Zbl 0714.16001. - Kanel-Belov, Alexei; Rowen, Louis Halle (2005).
*Computational aspects of polynomial identities*. Research Notes in Mathematics.**9**. Wellesley, MA: A K Peters. ISBN 1-56881-163-2. Zbl 1076.16018.