In mathematics, in the subfield of ring theory, a ringR is a polynomial identity ring if there is, for some N > 0, an element P other than 0 of the free algebra, Z, over the ring of integers in N variables X1, X2,..., XN such that for all N-tuples r1, r2,..., rN taken from Rit happens that 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 characteristicp 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 by 2 matrices over a commutative ring satisfies the Hall identity
A major role is played in the theory by the standard identitysN, of length N, which generalises the example given for commutative rings. It derives from the Leibniz formula for determinants
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 sN 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.
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 idealI of F is called T-ideal if for every endomorphismf 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.
