PI-rings

PI-rings
Name	PI-rings
Field	Ring theory, Algebra
Introduced	1940s–1950s
Properties	Noncommutative rings satisfying polynomial identities

PI-rings are associative rings that satisfy a nontrivial polynomial identity in noncommuting indeterminates. They occupy a central place in Ring theory and Noncommutative algebra and connect to the study of matrix algebras, division algebras, and polynomial identity algebras. PI-rings bridge classical results from Artin, Jacobson, and Amitsur to modern developments involving Kemer, Rowen, and Procesi.

Definition and Basic Properties

A PI-ring is an associative ring R (with unity or without depending on context of Kaplansky) for which there exists a nonzero polynomial f in noncommuting variables over a base ring (often field k) such that f vanishes under all substitutions by elements of R. Basic properties relate PI-rings to Noetherian, prime, semiprime, and simple rings: many structural theorems assert that prime PI-rings resemble matrix rings over commutative domains, while semiprime PI-rings decompose into subdirect products of prime PI-rings. Classical examples arise from matrix algebras like Mn(k) studied by Wedderburn, Burnside, and Schur.

Polynomial Identities and Examples

Typical polynomial identities include the standard polynomial S_n and the Capelli polynomial studied by Capelli and Razmyslov. Examples of PI-rings: full matrix algebras Mn(k) satisfy the standard identity S_{2n}, commutative algebras satisfy the commutator identity [x,y]=0, and upper triangular matrix algebras satisfy identities of smaller degree informed by work of Jacobson and Lewin. Group algebras k[G] for finite groups G often fail to be PI unless restrictions like Maschke conditions or group order constraints apply; enveloping algebras U(L) of finite-dimensional Lie algebras sometimes satisfy polynomial identities under conditions analyzed by Kaplansky and Posner.

Structure Theory and Representations

Structure theory for PI-rings draws on connections with Artinian and Goldie conditions, the Wedderburn–Artin structure of semisimple components, and representation theory through modules and identities. Results of Posner, Regev, and Small show how prime PI-rings embed into matrix algebras over division rings with centers that are finitely generated field extensions, aligning with concepts from central simple algebras and Brauer considerations. Representations of PI-rings often reduce to studying finite-dimensional representations of associated algebras, linking to Tits-type phenomena and to classification work of Dixmier in enveloping algebras.

PI-degree and Posner's Theorem

The PI-degree of a prime PI-ring is the minimal n such that the ring satisfies the standard identity S_{2n}. Posner's theorem asserts that a prime ring satisfying a polynomial identity is a primitive ring that is central over its center and embeds in a matrix algebra over a division algebra whose center is a field extension of the base field; this ties into Posner as developed alongside Amitsur and Kaplansky. The PI-degree is an invariant controlling growth, identities, and representation dimensions; it appears in theorems by Regev on codimension growth and in constraints from Small–Stafford–Warfield results on affine PI-algebras.

Amitsur–Levitzki and Standard Identities

The Amitsur–Levitzki theorem, proved by Amitsur and Levitzki, states that the full matrix algebra Mn(k) satisfies the standard polynomial S_{2n} and no nontrivial standard identity of lower degree. This foundational result links to work of Procesi on trace identities, Razmyslov on invariants, and to Formanek's studies of polynomial identities in free algebras. Standard identities, Capelli identities, and central polynomials provide tools to distinguish PI-classes and to construct T-ideals; these concepts interact with invariant theory explored by Weyl and Noether.

Varieties of PI-rings and T-ideals

The class of algebras satisfying a given set of polynomial identities forms a variety in the sense of universal algebra; T-ideals (ideals of polynomial identities in the free algebra) encode symbolic constraints. Kemer's breakthrough classification of T-ideals for affine PI-algebras over fields of characteristic zero connects to work of Kemer and to problems posed by Specht on finiteness of bases for identities. Varieties of PI-rings are studied via representability theorems, cocharacter sequences investigated by Berele and Drensky, and connections to combinatorial structures analyzed by Giambruno and Zaicev in asymptotic growth of codimensions.

Category:Ring theory