On graded characterizations of finite dimensionality for algebraic algebras

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.