Abstract
We observe that a finitely generated algebraic algebra R (over a field) is finite dimensional if and only if the associated graded ring \({{\text {gr}}}{R}\) is right noetherian, if and only if \({{\text {gr}}}{R}\) has right Krull dimension, if and only if \({{\text {gr}}}{R}\) satisfies a polynomial identity.
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I am grateful to Lance Small for helpful comments.
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Letzter, E.S. On graded characterizations of finite dimensionality for algebraic algebras. Arch. Math. 109, 499–503 (2017). https://doi.org/10.1007/s00013-017-1090-8
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DOI: https://doi.org/10.1007/s00013-017-1090-8