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".
Every commutative ring is a PI-ring, satisfying the polynomial identity XY − YX = 0.
To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.
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.
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.