Polynomial algebras have polynomial growth

Finston, David R.

doi:10.1090/S0002-9947-1987-0876465-6

Polynomial algebras have polynomial growth
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Trans. Amer. Math. Soc. 300 (1987), 535-556 Request permission

Abstract:

The definitions and basic properties of Gelfand-Kirillov dimension are extended to algebras over a field which are not necessarily associative. The results are applied to the algebra of polynomial functions on an arbitrary finite dimensional algebra to obtain polynomial growth (i.e. integral G-K dimension) for these algebras. The G-K dimension of the polynomial algebra in one indeterminate is shown to be constant on the category of all finite dimensional nomial extensions of an associative algebra.

References

Walter Borho and Hanspeter Kraft, Über die Gelfand-Kirillov-Dimension, Math. Ann. 220 (1976), no. 1, 1–24. MR 412240, DOI 10.1007/BF01354525
Carl Faith, Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR 0427349

The algebra of polynomial functions on a non-associative algebra

David R. Finston, Separable polynomials over finite-dimensional algebras, Comm. Algebra 13 (1985), no. 7, 1597–1626. MR 790400, DOI 10.1080/00927878508823241
David R. Finston, On multiplication algebras, Trans. Amer. Math. Soc. 293 (1986), no. 2, 807–818. MR 816327, DOI 10.1090/S0002-9947-1986-0816327-2
Nathan Jacobson, The Theory of Rings, American Mathematical Society Mathematical Surveys, Vol. II, American Mathematical Society, New York, 1943. MR 0008601
A. Joseph and L. W. Small, An additivity principle for Goldie rank, Israel J. Math. 31 (1978), no. 2, 105–114. MR 516246, DOI 10.1007/BF02760541
G. R. Krause and T. H. Lenagan, Growth of algebras and Gel′fand-Kirillov dimension, Research Notes in Mathematics, vol. 116, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 781129
Martin Lorenz and Lance W. Small, On the Gel′fand-Kirillov dimension of Noetherian P.I.-algebras, Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981) Contemp. Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1982, pp. 199–205. MR 685952
Helmut Röhrl, On the zeros of polynomials over arbitrary finite-dimensional algebras, Manuscripta Math. 25 (1978), no. 4, 359–390. MR 509591, DOI 10.1007/BF01168049

The algebra of polynomial functions with coefficients in a nonassociative algebra

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