= Solution
For $n\ge1$, the <Weyl algebra> $A=A_n(\mathbb C)$ is generated by $x_1,\ldots,x_n,D_1,\ldots,D_n$ with $[D_i,x_j]=\delta_{ij}$ and the other generator <commutators> zero. Iterating the differential <Ore extension> construction gives the ordered <basis> $x^\alpha D^\beta$. In the <Bernstein filtration>, both kinds of generator have degree one, and
$$
F_jA=\operatorname{span}\{x^\alpha D^\beta:|\alpha|+|\beta|\le j\},\qquad
\dim F_jA=\binom{j+2n}{2n},\qquad
\operatorname{gr}A\cong\mathbb C[x_1,\ldots,x_n,\xi_1,\ldots,\xi_n].
$$
Leading symbols show that $A$ is a <noncommutative domain>. The <Hilbert basis theorem> and homogeneous-generator lifting by decreasing degree make it both left and right <Noetherian>.
It is also a <simple ring>. In a nonzero two-sided <ideal>, choose a nonzero element and a <monomial> $x^{\alpha_0}D^{\beta_0}$ of maximal total degree with nonzero coefficient $c$. Applying $(\operatorname{ad}D)^{\alpha_0}(\operatorname{ad}x)^{\beta_0}$ eliminates every lower-degree term and every different term of that maximal degree. The result is the nonzero <scalar> $(-1)^{|\beta_0|}\alpha_0!\beta_0!c$. Thus the <ideal> contains $1$. Here the <scalar extraction by iterated Weyl commutators> uses characteristic zero. Despite simplicity, $A$ has no nonzero finite-dimensional <modules>: in such a representation $[D_1,x_1]=I$ would have <trace> $0=\dim M$, impossible over $\mathbb C$. Its interesting <module> theory is necessarily infinite-dimensional.
For a finitely generated left <module> $M$, choose a finite-dimensional generating subspace $M_0$ and set $M_j=F_jA\,M_0$. This is a <good filtration of a module>. Its <associated graded module> is finite over the graded polynomial <ring>, so its cumulative Hilbert function is eventually a polynomial,
$$
\dim M_j=\frac{e(M)}{d(M)!}j^{d(M)}+O(j^{d(M)-1})\quad(M\ne0),
$$
with positive integer multiplicity $e(M)$. These are the <Bernstein growth dimension of a Weyl algebra module> and its <multiplicity of a filtered module>. Two good filtrations bound each other after fixed shifts: lift finite homogeneous generators in one filtration, bound their degrees in the other, and use compatibility with $F_jA$; then reverse the roles. The growth degree and leading coefficient therefore do not depend on the <good filtration of a module>. The degree is also the <Gelfand–Kirillov dimension of a module> and the dimension of the <characteristic variety of a Weyl algebra module>, the support of $\operatorname{gr}M$. Since $M$ is a quotient of a finite sum of copies of $A$, $d(M)\le2n$.
The essential lower bound is the <Bernstein inequality for Weyl algebra modules>:
$$
\boxed{M\ne0\text{ finitely generated}\Longrightarrow d(M)\ge n.}
$$
Here is a full proof. Multiplication gives a linear map $F_jA\to\operatorname{Hom}_{\mathbb C}(M_j,M_{2j})$. Prove it injective by induction on $j$. At $j=0$ a <scalar> killing $M_0\ne0$ must vanish. If $a\in F_jA$ kills $M_j$, then for every generator $z$, its <commutator> $[a,z]\in F_{j-1}A$ kills $M_{j-1}$: both $azm$ and $zam$ are zero for $m\in M_{j-1}$. Induction gives $[a,z]=0$ for all generators. In the ordered <basis>, $[D_i,a]$ differentiates the $x_i$ exponents and $[x_i,a]$ differentiates the $D_i$ exponents with a minus sign. Vanishing of all these <commutators> in characteristic zero forces $a$ to be a <scalar>. It then vanishes because it kills $M_0$. This proves the <faithful finite-step action of a Weyl algebra> and hence
$$
\binom{j+2n}{2n}\le(\dim M_j)(\dim M_{2j})\le(\dim M_{2j})^2.
$$
Comparison of degrees gives $2n\le2d(M)$, proving the inequality. Equally, without invoking eventual polynomiality, this estimate directly bounds the <Gelfand–Kirillov dimension of a module> below by $n$. Right <modules> satisfy the same inequality through the filtration-preserving isomorphism $A\cong A^{\mathrm{op}}$, $x_i\mapsto x_i$, $D_i\mapsto-D_i$.
A nonzero finitely generated <module> with $d(M)=n$ is a <holonomic Weyl algebra module>; the zero <module> is conventionally included. This minimal-growth class is stable under <submodules>, <quotient modules> and extensions. For $0\to N\to M\to Q\to0$, take the induced and quotient filtrations. Their graded sequence is exact; <Noetherianity> of $\operatorname{gr}A$ makes the induced filtrations good. Thus the Hilbert functions add and
$$
d(M)=\max(d(N),d(Q));
$$
if both nonzero terms have the top dimension, their multiplicities add. For a <holonomic Weyl algebra module>, every nonzero subquotient has dimension $n$, by the upper bound inherited from $M$ and Bernstein's lower bound. Each nonzero factor in a strict chain therefore consumes positive integral multiplicity. This proves the <multiplicity bounds the length of a holonomic Weyl module>:
$$
\boxed{\operatorname{length}_A(M)\le e(M).}
$$
In particular a holonomic <module> is both <Noetherian> and <Artinian> and has finite length, although its vector-space dimension is infinite. This distinction separates finite-dimensional representation theory from finite-length Weyl <module> theory.
The polynomial <module> $\mathbb C[x_1,\ldots,x_n]$, with $x_i$ acting by multiplication and $D_i$ by differentiation, is generated by $1$. Its $M_j$ consists of polynomials of degree at most $j$, with dimension $\binom{j+n}{n}$, so $d=n$ and $e=1$. It is simple: differentiating a maximal-degree <monomial> of a nonzero polynomial isolates a nonzero constant, and multiplication by polynomials then generates the whole <module>. A second basic example is the delta <module>
$$
A\Big/\sum_iAx_i.
$$
Using the reverse ordered <basis> $D^\beta x^\alpha$, its <vector space> is $\mathbb C[D_1,\ldots,D_n]$; $D_i$ acts by multiplication and $x_i$ by negative differentiation. The same argument proves simplicity and holonomicity. These two examples are exchanged by the <automorphism> $x_i\mapsto D_i$, $D_i\mapsto-x_i$. Free <modules>, in contrast, have growth dimension $2n$ and are not holonomic. Thus the central organizing invariants are <good filtration of a module>, growth dimension, characteristic support and multiplicity, with Bernstein's estimate explaining why dimension $n$ is the natural smallest nonzero size.
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