Polynomial identity ring

In mathematics, in the subfield of ring theory, a ring R is a polynomial identity ring if there is, for some N > 0, an element P other than 0 of the free algebra, Z<X₁, X₂, ..., X_N>, over the ring of integers in N variables X₁, X₂, ..., X_N such that for all N-tuples r₁, r₂, ..., r_N taken from R it happens that

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.

Among noncommutative 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 ${\displaystyle p}$ is minimal over ${\displaystyle P\cap K}$) and the incomparability property (If P and Q are prime ideals of R and ${\displaystyle P\subset Q}$ then ${\displaystyle P\cap K\subset Q\cap K}$) are satisfied.

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