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$\langleX_{1}, X_{2}, ..., X_{N}\rangle$, over the ring of integers in N variables X1, X2, ..., XN such that
 * $$P(r_1, r_2, \ldots, r_N) = 0$$

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

 * For example, if R is a commutative ring it is a PI-ring: this is true with
 * $$P(X_1,X_2) = X_1X_2-X_2X_1 = 0~$$


 * The ring of 2&thinsp;×&thinsp;2 matrices over a commutative ring satisfies the Hall identity
 * $$(xy-yx)^2z=z(xy-yx)^2$$
 * This identity was used by, but was found earlier by.


 * A major role is played in the theory by the standard identity sN, of length N, which generalises the example given for commutative rings (N = 2). It derives from the Leibniz formula for determinants


 * $$\det(A) = \sum_{\sigma \in S_N} \sgn(\sigma) \prod_{i = 1}^N a_{i,\sigma(i)}$$


 * by replacing each product in the summand by the product of the Xi in the order given by the permutation σ. In other words each of the N&hairsp;! orders is summed, and the coefficient is 1 or −1 according to the signature.


 * $$s_N(X_1,\ldots,X_N) = \sum_{\sigma \in S_N} \sgn(\sigma) X_{\sigma(1)}\dotsm X_{\sigma(N)}=0~$$


 * The m&thinsp;×&thinsp;m matrix ring over any commutative ring satisfies a standard identity: the Amitsur–Levitzki theorem states that it satisfies s2m. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2m.


 * 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


 * ei&thinsp;ej = −&hairsp;ej&thinsp;ei.


 * This ring does not satisfy sN for any N and therefore can not be embedded in any matrix ring. In fact sN(e1,e2,...,eN) = N&hairsp;!&hairsp;e1e2...eN ≠ 0. On the other hand it is a PI-ring since it satisfies even degree commutes with every element. Therefore if either x or y is a monomial of even degree [x, y] := xy − yx = 0. If both are of odd degree then [x, y] = xy − yx = 2xy has even degree and therefore commutes with z, i.e. [[x, y], z] = 0.

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 sN for N >&thinsp;n and therefore it is a PI-ring.
 * If R and S are PI-rings then their tensor product over the integers, $$R\otimes_\mathbb{Z}S$$, is also a PI-ring.
 * If R is a PI-ring, then so is the ring of n&thinsp;×&thinsp;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 (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 $$P\cap K\subset Q\cap K$$) are satisfied.

The set of identities a PI-ring satisfies
If F := Z$\langleX_{1}, X_{2}, ..., X_{N}\rangle$ 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
 * $$\tau$$: F $$\rightarrow$$ R.

An ideal I of F is called T-ideal if $$f(I)\subset I$$ 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.