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, ..., XN⟩, over the ring of integers in N variables X1, X2, ..., XN such that for all N-tuples r1, r2, ..., rN taken from R. Strictly the Xi 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

Properties

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 ) are satisfied.

The set of identities a PI-ring satisfies

If F := Z⟨X1, X2, ..., XN⟩ 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.