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Books > Science & Mathematics > Mathematics > Algebra
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Rings with Polynomial Identities and Finite Dimensional Representations of Algebras (Paperback)
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Rings with Polynomial Identities and Finite Dimensional Representations of Algebras (Paperback)
Series: Colloquium Publications
Expected to ship within 12 - 17 working days
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A polynomial identity for an algebra (or a ring) $A$ is a
polynomial in noncommutative variables that vanishes under any
evaluation in $A$. An algebra satisfying a nontrivial polynomial
identity is called a PI algebra, and this is the main object of
study in this book, which can be used by graduate students and
researchers alike. The book is divided into four parts. Part 1
contains foundational material on representation theory and
noncommutative algebra. In addition to setting the stage for the
rest of the book, this part can be used for an introductory course
in noncommutative algebra. An expert reader may use Part 1 as
reference and start with the main topics in the remaining parts.
Part 2 discusses the combinatorial aspects of the theory, the
growth theorem, and Shirshov's bases. Here methods of
representation theory of the symmetric group play a major role.
Part 3 contains the main body of structure theorems for PI
algebras, theorems of Kaplansky and Posner, the theory of central
polynomials, M. Artin's theorem on Azumaya algebras, and the
geometric part on the variety of semisimple representations,
including the foundations of the theory of Cayley-Hamilton
algebras. Part 4 is devoted first to the proof of the theorem of
Razmyslov, Kemer, and Braun on the nilpotency of the nil radical
for finitely generated PI algebras over Noetherian rings, then to
the theory of Kemer and the Specht problem. Finally, the authors
discuss PI exponent and codimension growth. This part uses some
nontrivial analytic tools coming from probability theory. The
appendix presents the counterexamples of Golod and Shafarevich to
the Burnside problem.
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