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, over the ring of integers in "N" variables "X"1, "X"2, ..., "X""N" such that for all "N"-tuples "r"1, "r"2, ..., "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.
PI-rings as generalizations of commutative rings.
Among noncommutative rings, PI-rings satisfy the [[Köthe conjecture]]. [[Finitely generated algebra|Affine]] PI-algebras over a [[field (mathematics)|field]] satisfy the [[Kurosh conjecture]], the [[Nullstellensatz]] and the [[catenary ring|catenary property]] for [[prime ideals]].
If "R" is a PI-ring and "K" is a subring of its center such that "R" is [[integral extension|integral over]] "K" then the [[Going up and going down|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 formula_7 is minimal over formula_8) and the "incomparability" property (If "P" and "Q" are prime ideals of "R" and formula_9 then formula_10) are satisfied.
The set of identities a PI-ring satisfies.
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 (algebra)|kernel]] of any homomorphism
An ideal "I" of "F" is called T-ideal if formula_13 for every [[endomorphism]] "f" of "F".
Given a PI-ring, "R", the set of all polynomial identities it satisfies is an [[Ideal (ring theory)|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.
External links.
[[Category:Ring theory]]