# On graded characterizations of finite dimensionality for algebraic algebras

## TL;DR

This paper establishes that for finitely generated algebraic algebras over a field, finite dimensionality is equivalent to their associated graded rings being right noetherian, having right Krull dimension, or satisfying a polynomial identity.

## Contribution

It provides a new characterization of finite dimensionality for algebraic algebras via properties of their associated graded rings.

## Key findings

- Finite dimensionality is equivalent to the associated graded ring being right noetherian.
- Finite dimensionality is equivalent to the associated graded ring having right Krull dimension.
- Finite dimensionality is equivalent to the associated graded ring satisfying a polynomial identity.

## Abstract

We observe that a finitely generated algebraic algebra R (over a field) is finite dimensional if and only if the associated graded ring grR is right noetherian, if and only if grR has right Krull dimension, if and only if grR satisfies a polynomial identity.