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The Goldie conditions for Algebras with Bounded Growth

The Goldie conditions for Algebras with Bounded Growth THE GOLDIE CONDITIONS FOR ALGEBRAS WITH BOUNDED GROWTH RONALD S. IRVING AND LANCE W. SMALL 1. Introduction In this paper we consider possible generalizations of results of Jategaonkar and Bergman [6, 7] which state that a domain with no free subalgebras, or subexponential growth, has a division ring of fractions. Recall for an algebra A generated over a field K by a finite set X that the growth function of A with respect to X is the function f :N -+ N with f(n) equal to the dimension of the space spanned by all monomials on X of length ^ n. If / is bounded by a polynomial, A is said to have polynomially bounded growth, and if / grows exponentially, A has 1/ n exponential growth. More precisely, A has exponential growth if lim/(n) > 1; otherwise A has subexponential growth. In case A has polynomially bounded growth, the infimum of degrees of functions n bounding / is the Gelfand-Kirillov dimension of A; it is independent of the choice of X. For algebras A which are not finitely-generated, these terms still apply via a consideration of all finitely-generated subalgebras. The striking results of Jategaonkar and Bergman suggest that http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

The Goldie conditions for Algebras with Bounded Growth

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/15.6.596
Publisher site
See Article on Publisher Site

Abstract

THE GOLDIE CONDITIONS FOR ALGEBRAS WITH BOUNDED GROWTH RONALD S. IRVING AND LANCE W. SMALL 1. Introduction In this paper we consider possible generalizations of results of Jategaonkar and Bergman [6, 7] which state that a domain with no free subalgebras, or subexponential growth, has a division ring of fractions. Recall for an algebra A generated over a field K by a finite set X that the growth function of A with respect to X is the function f :N -+ N with f(n) equal to the dimension of the space spanned by all monomials on X of length ^ n. If / is bounded by a polynomial, A is said to have polynomially bounded growth, and if / grows exponentially, A has 1/ n exponential growth. More precisely, A has exponential growth if lim/(n) > 1; otherwise A has subexponential growth. In case A has polynomially bounded growth, the infimum of degrees of functions n bounding / is the Gelfand-Kirillov dimension of A; it is independent of the choice of X. For algebras A which are not finitely-generated, these terms still apply via a consideration of all finitely-generated subalgebras. The striking results of Jategaonkar and Bergman suggest that

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1983

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