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⟨X₁, X₂, ..., X_N⟩, over the ring of integers in N variables X₁, X₂, ..., X_N such that

P(r_{1},r_{2},\ldots ,r_{N})=0

for all N-tuples r₁, r₂, ..., 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.^[1]

Examples

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

P(X_{1},X_{2})=X_{1}X_{2}-X_{2}X_{1}=0~

The ring of 2 × 2 matrices over a commutative ring satisfies the Hall identity

(xy-yx)^{2}z=z(xy-yx)^{2}

This identity was used by M. Hall (1943), but was found earlier by Wagner (1937).

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 _{\sigma \in S_{N}}\operatorname {sgn}(\sigma )\prod _{i=1}^{N}a_{i,\sigma (i)}

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 _{\sigma \in S_{N}}\operatorname {sgn}(\sigma )X_{\sigma (1)}\dotsm X_{\sigma (N)}=0~

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 2m.

Given a field k of characteristic zero, take R to be the exterior algebra over a countably infinite-dimensional vector space with basis e₁, e₂, e₃, ... 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(e₁,e₂,...,e_N) = N ! e₁e₂...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 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 = 2xy 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, $R\otimes _{\mathbb {Z} }S$ , 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⟨X₁, X₂, ..., 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

\tau

: F

\rightarrow

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.

References

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

Latyshev, V.N. (2001) [1994], "PI-algebra", Encyclopedia of Mathematics, EMS Press
Formanek, E. (2001) [1994], "Amitsur–Levitzki theorem", Encyclopedia of Mathematics, EMS Press
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