Polynomial identity ring

- The ring of 2 × 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** *s**N*, 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 *X**i* in the order given by the permutation σ. In other words each of the *N*! 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* × *m* matrix ring over any commutative ring satisfies a standard identity: the Amitsur–Levitzki theorem states that it satisfies *s*2*m*. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2*m*. - Given a field *k* of characteristic zero, take *R* to be the exterior algebra over a countably infinite-dimensional vector space with basis *e*1, *e*2, *e*3, ... Then *R* is generated by the elements of this basis and ::*e**i* *e**j* = −*e**j* *e**i*. :This ring does not satisfy *s**N* for any *N* and therefore can not be embedded in any matrix ring. In fact *s**N*(*e*1,*e*2,...,*e**N*) = *N*!*e*1*e*2...*e**N* ≠ 0. On the other hand it is a PI-ring since it satisfies [[*x*, *y*], *z*] := *xyz* − *yxz* − *zxy* + *zyx* = 0. It is enough to check this for monomials in the *e**i*'s. Now, a monomial of 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* = 2*xy* 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 *s**N* for *N* *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* × *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** 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. ## References - - - [Polynomial identities in ring theory](https://books.google.com/books?id=Li147JZ4T6AC), Louis Halle Rowen, Academic Press, 1980, - [Polynomial identity rings](https://books.google.com/books?id=x8gJw5bMw2oC), Vesselin S. Drensky, Edward Formanek, Birkhäuser, 2004, - [Polynomial identities and asymptotic methods](https://books.google.com/books?id=ZLW_Kz_zOP8C), A. Giambruno, Mikhail Zaicev, AMS Bookstore, 2005, - [Computational aspects of polynomial identities](https://books.google.com/books?id=80pw1QoLSQUC), Alexei Kanel-Belov, Louis Halle Rowen, A K Peters Ltd., 2005, ## References 1. J.C. McConnell, J.C. Robson, ''Noncommutative Noetherian Rings, [[Graduate Studies in Mathematics]], Vol 30'' ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Polynomial_identity_ring) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Polynomial_identity_ring?action=history). ::