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 X₁, X₂,..., X_N such that
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.

Examples

For example, if R is a commutative ring it is a PI-ring: this is true with
The ring of 2 × 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. It derives from the Leibniz formula for determinants
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

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 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 and the incomparability property 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
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.