Book Review: Computational aspects of polynomial identities
Edward Formanek · Bulletin of the American Mathematical Society · 2006
Let k be an infinite field, and let k X = k x 1 , x 2 , . . .be a free associative algebra over k in countably many variables.If f (x 1 , . . ., x n ) ∈ k X and r 1 , . . ., r n are elements of a k-algebra A, then one can make the evaluation f (r 1 , . . ., r n ) ∈ A. If all such evaluations on A are zero, then f is called a polynomial identity for A. The zero polynomial is a polynomial identity for any k-algebra A, and it may be the only polynomial identity for A. If some nonzero f is a polynomial identity for A, then one says that A is a polynomial identity algebra over k or, more simply, a polynomial identity ring, or PI-ring.Examples of PI-rings are commutative rings, matrix rings over commutative rings, finite-dimensional algebras, algebraic algebras of bounded degree, and exterior algebras.Commutative rings satisfy the polynomial x 1 x 2x 2 x 1 , and "satisfying a polynomial identity" is a generalization of commutativity.The set of f that vanish on A forms an ideal T (A) in k X called the T-ideal of identities of A. It is easy to see that the T-ideals of k X are precisely those ideals which are closed under k-endomorphisms of k X .Polynomial identities are also defined for algebras over arbitrary commutative rings, but this involves a minor technicality which is best avoided in a review.The theory of polynomial identity rings has two branches, a ring-theoretic or structural and a combinatorial or computational, both of which have well-defined sources.The former, which began with a paper of I. Kaplansky [13] in 1948, studies the ring-theoretic properties of polynomial identity rings.The latter, which began with a paper of W. Specht [31] in 1950, studies T-ideals.A handful of significant earlier papers by M. Dehn [6] in 1922, W. Wagner [32] in 1937, and M. Hall [12] in 1943 involved polynomial identities, but it did not become a well-defined field until 1948.Indeed, the term "polynomial identity" was introduced by Kaplansky, and "T-ideal" was introduced by Specht.Kaplansky not only began the field of polynomial identities, but he proved the single most important PI-theorem: A primitive PI-ring is a finite-dimensional central simple algebra over its center, which is a field.Kaplansky's Theorem is the precursor to two other major structure theorems, Posner's Theorem and Artin's Theorem, both of which retain the PI-hypothesis and replace the hypothesis of primitivity with a less restrictive one.Specht's paper was rather formal and did not contain any groundbreaking result like Kaplansky's Theorem, but it did provide the foundation for further research in the quantitative side of PI-theory.He assumed that the base field k had characteristic zero, which has two advantages.First, over a field of characteristic zero any T-ideal is generated by the multilinear polynomials it contains.Second, the multilinear polynomials in x 1 , . . ., x n are a module over S n , the symmetric group on n letters, which means that the highly developed representation theory of S n in characteristic zero is available.He also posed Specht's Problem: If k has characteristic zero, is every T -ideal in k X finitely generated as a T -ideal?(For J to be generated as a T -ideal by a set A means that J is 2000 Mathematics Subject Classification.Primary 16R10.