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For a finitely generated algebra over $k$, choose a finite-dimensional generating subspace $V$ containing one. Its Gelfand–Kirillov dimension isChanging generators bounds the two filtrations by constant rescalings of the index, so the value is independent of $V$.

For the noncommutative Krull dimension, set the zero module's dimension to $-1$. A nonzero module has dimension zero exactly when it is an Artinian module. Recursively, dimension at most $d$ means that in every descending chain of submodules the successive factors eventually have dimension less than $d$; use the ordinal version of this definition if necessary. The left Krull dimension of $R$ is that of its left regular module. For commutative Noetherian rings this agrees with the usual prime-chain Krull dimension, but prime-chain counting alone is not its definition for a noncommutative ring.

Write $U=U({\mathfrak{sl}}_2(\mathbb{C}))$ with $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$. By the Poincaré-Birkhoff-Witt theorem, $f^i{h}^j{e}^k$ form a basis. In the total-degree filtration there are $\left(\genfrac{}{}{0ex}{}{n+3}{3}\right)$ such monomials of degree at most $n$, soThe commutators lower degree, giving $\mathrm{gr}U\cong \mathbb{C}[f,h,e]$, which also proves that $U$ is left and right Noetherian by finite leading-term cancellation.

To determine the module-theoretic Krull dimension, use the generalized Weyl algebra description and verify the dimension criterion, rather than substituting the three-dimensional graded ring. PutCommutation with $h$ is immediate; with $e$, the contributions are $4he-2(eh+he)-4e=0$, using $he-eh=2e$. The corresponding calculation with $f$ also vanishes. Thus $z$ is a Casimir element. The subalgebra $D=\mathbb{C}[h,z]$ is a polynomial ring in two variables: for a nonzero polynomial in $z$ with coefficients in $\mathbb{C}[h]$, its highest $z$ power contributes a nonzero PBW term with that many copies of both $f$ and $e$, and no lower power can cancel it.

Take $\sigma (h)=h-2$, $\sigma (z)=z$ and $t=(z-h^2-2h)/4$. The relations becomeThey identify $U$ with $D(\sigma ,t)$. Surjectivity follows from its generators; injectivity follows from PBW: the graded pieces $D$, $D{e}^r$ and $D{f}^r$ for $r>0$ are linearly independent, using the difference between the $e$ and $f$ exponents and then the highest paired exponent. Every PBW monomial lies in one of these pieces by replacing paired $f,e$ powers with polynomials in $h,z$.

We use the following general Krull dimension criterion for rank-one generalized Weyl algebras: for a commutative Noetherian base $D$ of dimension $d$, the generalized Weyl algebra has Krull dimension $d$ provided no height-$d$ maximal ideal is periodic under $\sigma$ and none contains infinitely many translates ${\sigma }^n(t)$. This is Theorem 5.3, specialized to the commutative base, in Krull dimension of Generalized Weyl Algebras with non-commutative coefficients.

Here every maximal ideal is ${\mathfrak{m}}_{\lambda ,\mu }=(h-\lambda ,z-\mu )$ and has height two. Its translates have $h$ coordinate $\lambda +2n$, so no nonzero power of $\sigma$ fixes it. Further,This is a nonzero quadratic polynomial in $n$, so it vanishes for at most two integers. Both exceptional mechanisms are excluded for every maximal ideal, and the base has dimension two. The criterion consequently givesThe orbit and root calculations are needed for this upper bound on the entire left-ideal lattice; merely exhibiting a length-two chain of prime ideals would only prove a lower bound.