Throughout is a field. Let be a -algebra. Let be a -vector subspace of spanned by the set For any integer we will denote by the the -subspace of spanned by all monomials of length in We will also define If then is called a **generating subspace** of If a generating subspace of contains 1, then is called a **frame** of We will denote by the subspace of Clearly if is a generating subspace of then

**Remark 1**. Let be a finitely generated -algebra and suppose that and are two generating subspaces of Then

*Proof*. We have Since both and are finite dimensional, there exist integers and such that and Thus and for all integers Now, implies that

Taking limsup of both sides of the above inequality will give us

because and

Similarly will imply

which completes the proof.

So, by the above remark, the value does not depend on the generaing subspace and thus the following definition makes sense.

**Definition**. Let be a finitely generated -algebra and let be a generating subspace of The **Gelfand-Kirillov dimension**, or GK dimension, of which we will denote it by is defined by

The GK dimension of a algebra somehow measures how far an algebra is from being finite dimensional. We will see later that if is commutative, then the GK dimension of is nothing but the transcendence degree of

**Remark 2**. In the above definition if is a frame of then because

Thus we get this simple expression

Hey Yaghoub. Maybe it’s because I haven’t read enough of the after-this GK dimension posts, but what precisely is the point of it? I mean, I get roughly what it measures, but is there a typical place that GK dimension would come up?

Yes, GK dimension is a useful and important invariant of algebras, although it’s a relatively new subject in ring theory.

For example, if k is an algebraically closed filed and A is a finitely generated k-algebra as well as a domain, then GKdim(A) = 1 implies that A is commutative. This is an important result.

For commutative algebras GK dimension is not very interesting because it coincides with Krul dimension, which is a much more known invariant.

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Thank you! Do you do work with GK dimension?

Not directly. I’m working on the ring structure of the centralizer of an element in an algebra and GK dimension is a useful tool in my research.