A lead figure in twentieth century noncommutative algebra, S. A. Amitsur's contributions are wide-ranging and enduring. This volume collects almost all of his work. The papers are organized into broad topic areas: general ring theory, rings satisfying a polynomial identity, combinatorial polynomial identity theory, and division algebras. Included are essays by the editors on Amitsur's work in these four areas and a biography of Amitsur written by A. Mann. This volume makes a fine addition to any mathematics book collection.

Readership

Graduate students and research mathematicians interested in ring theory, combinatorial algebra, and number theory.

Reviews

"These volumes are a collection of short and very readable papers, which many working algebraists will want to own; they should certainly form part of every mathematics library."

-- Zentralblatt MATH

Table of Contents

• Amitsur and ring theory
• A generalization of a theorem on linear differential equations
• A general theory of radicals. I. Radicals in complete lattices
• A general theory of radicals. II. Radicals in rings and bicategories
• A general theory of radicals. III. Applications
• Algebras over infinite fields
• Radicals of polynomial rings
• Invariant submodules of simple rings
• Derivations in simple rings
• The radical of field extensions
• Countably generated division algebras over nondenumerable fields
• Commutative linear differential operators
• Rings with a pivotal monomial
• On the semi-simplicity of group algebras
• Derived functors in abelian categories
• Remarks on principal ideal rings
• Generalized polynomial identities and pivotal monomials
• Rings with involution
• Rings of quotients and Morita contexts
• Nil radicals. Historical notes and some new results
• On rings of quotients
• Recognition of matrix rings II

• Amitsur and PI rings
• Nil PI-rings
• An embedding of PI-rings
• On rings with identities
• The \(T\)-ideals of the free ring
• A generalization of Hilbert's Nullstellensatz
• Groups with representations of bounded degree II
• Jacobson-rings and Hilbert algebras with polynomial identities
• Nil semi-groups of rings with a polynomial identity
• Rational identities and applications to algebra and geometry
• Prime rings having polynomial identities with arbitrary coefficients
• Identities in rings with involutions
• A noncommutative Hilbert basis theorem and subrings of matrices
• Embeddings in matrix rings
• Some results on rings with polynomial identities
• A note on PI-rings
• On universal embeddings in matrix rings
• Polynomial identities and Azumaya algebras
• Polynomial identities
• Central embeddings in semi-simple rings
• Polynomials over division rings
• Prime ideals in PI-rings
• Finite-dimensional representations of PI algebras
• GK-dimensions of corners and ideals
• Contributions of PI theory to Azumaya algebras
• Finite-dimensional representation of PI algebras, II
• Algebras over infinite fields, revisited
• Acknowledgement

• Amitsur and combinatorial P.I. theory
• Minimal identities for algebras
• Remarks on minimal identities for algebras
• The identities of PI-rings
• Identities and generators of matrix rings
• Identities and linear dependence
• On a central identity for matrix rings
• Alternating identities
• PI-algebras and their cocharacters
• The sequence of codimensions of PI-algebras

• Amitsur and division algebras
• Contributions to the theory of central simple algebras
• La représentation d'algèbres centrales simples
• Construction d'algèbres centrales simples sur des corps de caractéristique zéro
• Non-commutative cyclic fields
• Differential polynomials and division algebras
• Generic splitting fields of central simple algebras
• Finite subgroups of division rings
• Some results on central simple algebras
• On arithmetic functions
• Simple algebras and cohomology groups of arbitrary fields
• Some results on arithmetic functions
• Finite dimensional central divison algebras
• Homology groups and double complexes for arbitrary fields
• On a lemma in elementary proofs of the prime number theorem
• Complexes of rings
• On central division algebras
• The generic division rings
• Generic abelian crossed products and \(p\)-algebras
• Division algebras of degree 4 and 8 with involution
• On the characteristic polynomial of a sum of matrices
• Generic splitting fields. Brauer groups in ring theory and algebraic geometry.
• Extension of derivations to central simple algebras
• Kummer subfields of Malcev-Neumann division algebras
• Symplectic modules
• Totally ramified splitting fields of central simple algebras over Henselian fields
• Galois splitting fields of a universal division algebra
• Elements of reduced trace 0
• Finite-dimensional subalgebras of division rings
• Acknowledgment