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

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

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]