A comprehensive study of the main research done in polynomial identities over the last 25 years, including Kemer's solution to the Specht problem in characteristic O and examples in the characteristic p situation. The authors also cover codimension theory, starting with Regev's theorem and continuing through the Giambruno-Zaicev exponential rank. The "best" proofs of classical results, such as the existence of central polynomials, the tensor product theorem, the nilpotence of the radical of an affine PI-algebra, Shirshov's theorem, and characterization of group algebras with PI, are presented.

Computational Aspects of Polynomial Identities: Volume l, Kemer's Theorems, 2nd Edition presents the underlying ideas in recent polynomial identity (PI)-theory and demonstrates the validity of the proofs of PI-theorems. This edition gives all the details involved in Kemer's proof of Specht's conjecture for affine PI-algebras in characteristic 0. The book first discusses the theory needed for Kemer's proof, including the featured role of Grassmann algebra and the translation to superalgebras. The authors develop Kemer polynomials for arbitrary varieties as tools for proving diverse theorems. They also lay the groundwork for analogous theorems that have recently been proved for Lie algebras and alternative algebras. They then describe counterexamples to Specht's conjecture in characteristic p as well as the underlying theory. The book also covers Noetherian PI-algebras, Poincare-Hilbert series, Gelfand-Kirillov dimension, the combinatoric theory of affine PI-algebras, and homogeneous identities in terms of the representation theory of the general linear group GL. Through the theory of Kemer polynomials, this edition shows that the techniques of finite dimensional algebras are available for all affine PI-algebras. It also emphasizes the Grassmann algebra as a recurring theme, including in Rosset's proof of the Amitsur-Levitzki theorem, a simple example of a finitely based T-ideal, the link between algebras and superalgebras, and a test algebra for counterexamples in characteristic p.

Some of the important advances in Polynomial Identity (PI) theory in the last twenty years have remained accessible only to experts, limiting the exposure of advanced aspects of PI-theory to the general mathematical community. This book's main objective is to describe these breakthroughs in full, starting with Shirshov's theorem, discussing Kemer's solution of Specht's conjecture in characteristic zero, and completing with a proof of the theorem. The authors detail the theory needed for this proof in the early chapters of the book. Later chapters discuss related topics such as counterexamples to Specht's conjecture in characteristic p, Noetherian PI-algebras, Poincare-Hilbert series, Gelfand-Kirillov dimension, the combinatoric theory of affine PI-algebras, the ideals of identities, multilinear identities in terms of representation theory, and trace identities.

Computational Aspects of Polynomial Identities: Volume l, Kemer's Theorems, 2nd Edition presents the underlying ideas in recent polynomial identity (PI)-theory and demonstrates the validity of the proofs of PI-theorems. This edition gives all the details involved in Kemer's proof of Specht's conjecture for affine PI-algebras in characteristic 0. The book first discusses the theory needed for Kemer's proof, including the featured role of Grassmann algebra and the translation to superalgebras. The authors develop Kemer polynomials for arbitrary varieties as tools for proving diverse theorems. They also lay the groundwork for analogous theorems that have recently been proved for Lie algebras and alternative algebras. They then describe counterexamples to Specht's conjecture in characteristic p as well as the underlying theory. The book also covers Noetherian PI-algebras, Poincare-Hilbert series, Gelfand-Kirillov dimension, the combinatoric theory of affine PI-algebras, and homogeneous identities in terms of the representation theory of the general linear group GL. Through the theory of Kemer polynomials, this edition shows that the techniques of finite dimensional algebras are available for all affine PI-algebras. It also emphasizes the Grassmann algebra as a recurring theme, including in Rosset's proof of the Amitsur-Levitzki theorem, a simple example of a finitely based T-ideal, the link between algebras and superalgebras, and a test algebra for counterexamples in characteristic p.

This book is a companion volume to Graduate Algebra: Commutative View (published as volume 73 in this series). The main and most important feature of the book is that it presents a unified approach to many important topics, such as group theory, ring theory, Lie algebras, and gives conceptual proofs of many basic results of noncommutative algebra. There are also a number of major results in noncommutative algebra that are usually found only in technical works, such as Zelmanov's proof of the restricted Burnside problem in group theory, word problems in groups, Tits's alternative in algebraic groups, PI algebras, and many of the roles that Coxeter diagrams play in algebra.