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AuthorDrensky, Vesselin. author
TitlePolynomial Identity Rings [electronic resource] / by Vesselin Drensky, Edward Formanek
ImprintBasel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2004
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Descript VII, 200 p. online resource


A ring R satisfies a polynomial identity if there is a polynomial f in noncommuting variables which vanishes under substitutions from R. For example, commutative rings satisfy the polynomial f(x,y) = xy - yx and exterior algebras satisfy the polynomial f(x,y,z) = (xy - yx)z - z(xy - yx). "Satisfying a polynomial identity" is often regarded as a generalization of commutativity. These lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The former studies the ideal of polynomial identities satisfied by a ring R. The latter studies the properties of rings which satisfy a polynomial identity. The greater part of recent research in polynomial identity rings is about combinatorial questions, and the combinatorial part of the lecture notes gives an up-to-date account of recent research. On the other hand, the main structural results have been known for some time, and the emphasis there is on a presentation accessible to newcomers to the subject. The intended audience is graduate students in algebra, and researchers in algebra, combinatorics and invariant theory


A Combinatorial Aspects in PI-Rings -- Vesselin Drensky -- 1 Basic Properties of PI-algebras -- 2 Quantitative Approach to PI-algebras -- 3 The Amitsur-Levitzki Theorem -- 4 Central Polynomials for Matrices -- 5 Invariant Theory of Matrices -- 6 The Nagata-Higman Theorem -- 7 The Shirshov Theorem for Finitely Generated PI-algebras -- 8 Growth of Codimensions of PI-algebras -- B Polynomial Identity Rings -- Edward Formanek -- 1 Polynomial Identities -- 2 The Amitsur-Levitzki Theorem -- 3 Central Polynomials -- 4 Kaplanskyโ{128}{153}s Theorem -- 5 Theorems of Amitsur and Levitzki on Radicals -- 6 Posnerโ{128}{153}s Theorem -- 7 Every PI-ring Satisfies a Power of the Standard Identity -- 8 Azumaya Algebras -- 9 Artinโ{128}{153}s Theorem -- 10 Chain Conditions -- 11 Hilbert and Jacobson PI-Rings -- 12 The Ring of Generic Matrices -- 13 The Generic Division Ring of Two 2 x 2 Generic Matrices -- 14 The Center of the Generic Division Ring -- 15 Is the Center of the Generic Division Ring a Rational Function Field?

Mathematics Associative rings Rings (Algebra) Combinatorics Mathematics Associative Rings and Algebras Combinatorics


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