Simplicity of the Weyl Algebra

Q: Why does simplicity not make A_n a field or a division ring?

Because that implication needs commutativity. A commutative simple ring is a field, since for a ≠ 0 the ideal (a) is everything, so a has an inverse. In the noncommutative case the two-sided ideal generated by a is ∑ b_i a c_i , and getting 1 out of that says nothing about a having an inverse. For a = x in A_1 we have ∂ x - x∂ = 1 , which certainly does not make x invertible.

Q: Where does the proof use that K has characteristic zero?

Exactly once, in the assertion that β_i c_αβ ≠ 0 when β_i ≠ 0 as an integer. In characteristic p the integer β_i can be zero in K , the bracket vanishes, and the descent stops. This is not a defect of the proof: the theorem itself is false in characteristic p .

Q: Is the descent effective - can I actually compute the constant?

Yes, and formula (2.15) gives it in closed form: pick any top-degree pair (α,β) with c_αβ ≠ 0 , apply ad,∂ and -ad,x the corresponding number of times, and the answer is α!,β!,c_αβ . The worked example carries this out and the predicted value 4 is confirmed by direct computation.

Q: Does simplicity imply that A_n has a unique simple module?

No, and this is a common confusion. A simple ring has trivial two-sided ideal structure; a simple module is one with no proper non-zero submodules. Simple artinian rings do have a unique simple module up to isomorphism, but A_n is not artinian and has a rich supply of non-isomorphic simple modules - for instance K[x] and the twisted modules A_1/A_1(x∂-λ) for suitable λ .

Q: If every non-zero module is faithful, why is that useful?

Because it turns a statement about the size of the algebra into a statement about the size of the module . The whole of A_n must act by distinct operators on M , and _K of the degree- m part of A_n grows like m^2n . Comparing that growth to the growth of M is precisely the proof of Bernstein's inequality.

Q: What is the two-sided ideal structure in characteristic p?

Rich. The centre of the generators-and-relations Weyl algebra over Z_p is the polynomial ring Z_p[x_1^p, … ,x_n^p,∂_1^p, … ,∂_n^p] , and the algebra is a free module of rank p^2n over it. Every ideal of the centre generates a two-sided ideal, so there are as many as in a polynomial ring in 2n variables. Simplicity fails in the strongest possible way.

Q: Is there a version of simplicity for the ring B_n(K) of operators with rational coefficients?

Yes. B_n(K) , the K(x_1, … ,x_n) -algebra generated by the ∂_i , is also a simple domain in characteristic zero, and Coutinho develops this in the exercises to Ch. 2 using the order rather than the degree. For n = 1 it has the additional feature of a left division algorithm, so every left ideal is principal - a property A_1 does not have.

Overview

A ring is simple when its only two-sided ideals are the zero ideal and the ring itself. Commutative simple rings are exactly the fields, so simplicity is a very restrictive condition in the commutative world. Noncommutative rings behave differently, and the Weyl algebra is the standard illustration: $A_n$ is simple, yet it is a domain whose only units are the non-zero scalars. It is simple and nowhere near being a division ring.

The proof takes a non-zero two-sided ideal $I$ , picks an element $D ∈ I$ of least degree, and shows that if $(D) > 0$ then bracketing $D$ with a suitable generator produces a non-zero element of $I$ of strictly smaller degree. Since degrees are non-negative integers this cannot go on forever, so the minimal degree is $0$ , that is $I$ contains a non-zero constant, hence $I = A_n$ .

The single point where the argument can fail is the word "non-zero". Bracketing with $∂_i$ differentiates the coefficient polynomial with respect to $x_i$ ; bracketing with $x_i$ differentiates it with respect to the symbol variable $ξ_i$ . In characteristic zero, differentiating a non-constant polynomial in one of its variables never gives zero in all of them at once. In characteristic $p$ it does - $∂(x^p)/∂ x = 0$ - and simplicity genuinely fails there. That is the reason for the standing hypothesis on $K$ .

Two consequences carry a lot of weight later. Every $K$ -algebra endomorphism of $A_n$ is injective, because its kernel is a two-sided ideal not containing $1$ . And every non-zero $A_n$ -module is faithful, which is exactly the leverage used in the proof of Bernstein's inequality.

Definition

Throughout, $K$ is a field of characteristic zero and $A_n = A_n(K)$ .

Simple ring

A ring $R$ with $1 ≠ 0$ is simple if its only two-sided ideals are $0$ and $R$ . Equivalently, for every non-zero $a ∈ R$ the two-sided ideal $R a R$ generated by $a$ is all of $R$ : there exist finitely many $b_i, c_i ∈ R$ with $∑ _i b_i a c_i = 1$ .

The Weyl algebra is simpleCoutinho (2.2.1)

Let $K$ be a field of characteristic zero and $n ≥ 1$ . Then $A_n(K)$ is a simple ring.

Simple does not mean "few ideals"

Simplicity is a statement about two-sided ideals only. The left ideal $A_1∂$ is proper and non-zero, and so is $A_1 P$ for every non-constant $P$ . Indeed the whole of $D$ -module theory is the study of the left ideals of $A_n$ and the modules they define. The theorem says only that no proper non-zero left ideal is closed under right multiplication as well.

Core Concepts

The mechanism is easiest to see through a single translation.

Bracketing is differentiation of the coefficient polynomial

Write $D ∈ A_n$ in canonical form $D = ∑ c_αβx^α∂^β$ and encode it as the commutative polynomial $P_D(x,ξ) = ∑ c_αβx^αξ^β$ - not just its top-degree part, the whole thing. Then the two adjoint actions are exactly partial differentiation:

$[∂_i, D]$ corresponds to $∂ P_D/∂ x_i$ , and $[x_i, D]$ corresponds to $-,∂ P_D/∂ ξ_i$ . These identities are exact on canonical monomials, with no lower-order corrections, because $[∂_i, x^α∂^β] = α_i x^α-e_i∂^β$ and $[x_i, x^α∂^β] = -β_i x^α∂^β-e_i$ .

A two-sided ideal is closed under all $2n$ derivatives

If $I$ is a two-sided ideal and $D ∈ I$ , then $[y, D] = yD - Dy ∈ I$ for every $y ∈ A_n$ . Taking $y$ among the $2n$ generators, $I$ corresponds to a subspace of $K[x,ξ]$ closed under all $2n$ partial derivatives. In characteristic zero the only such non-zero subspaces contain the constants: differentiate a non-zero polynomial enough times in the right variables and you land on a non-zero scalar.

That is the whole theorem

A non-zero constant in $I$ is a unit, so $I = A_n$ . In characteristic $p$ the statement "a subspace of $K[x,ξ]$ closed under all partial derivatives and containing a non-zero element contains a non-zero constant" is false: the span of $x_1^p$ is closed under every partial derivative and contains no constant. That single sentence explains both the theorem and its failure.

Construction and Proof

The bracket formulas

For all $i$ and all multi-indices $α, β$ , identities (2.12) and (2.13) hold exactly.

Proof of the lemma

Since $∂_i$ commutes with every $∂_j$ , the derivation identity $[a,bc] = [a,b]c + b[a,c]$ gives $[∂_i, x^α∂^β] = [∂_i, x^α]∂^β$ . And $[∂_i, f] = ∂ f/∂ x_i$ for any polynomial $f$ , applied to $f = x^α$ , gives $α_i x^α-e_i$ . That is (2.12).

Symmetrically, $x_i$ commutes with every $x_j$ , so $[x_i, x^α∂^β] = x^α[x_i,∂^β] = -x^α[∂^β, x_i]$ . Induction on $|β|$ using $[∂_j, x_i] = δ_ij$ gives $[∂^β, x_i] = β_i∂^β - e_i$ , which is (2.13). Note both results are again canonical monomials, so no normalisation is needed afterwards - this is what makes the descent bookkeeping exact.

Proof that $A_n$ is simpleCoutinho (2.2.1)

Let $I ≠ 0$ be a two-sided ideal and choose $D ∈ I$ , $D ≠ 0$ , with $k = (D)$ minimal among non-zero elements of $I$ .

Suppose first $k = 0$ . Then $D$ is a non-zero element of $K$ , hence a unit, so $I = A_n$ and we are done.

Suppose $k > 0$ , aiming for a contradiction. Some coefficient $c_αβ$ of $D$ with $|α| + |β| = k$ is non-zero. Because $k > 0$ , either $β ≠ 0$ or $α ≠ 0$ .

If $β_i ≠ 0$ for some $i$ , consider $[x_i, D] ∈ I$ . By (2.13) its coefficient polynomial is $-∂ P_D/∂ ξ_i$ , whose coefficient of $x^αξ^β - e_i$ is $-β_i c_αβ$ . In characteristic zero $β_i ≠ 0$ in $K$ , so this is non-zero and $[x_i, D] ≠ 0$ . By (2.13) again its degree is at most $k - 1$ . This contradicts minimality.

Otherwise $β = 0$ , and since $k > 0$ some $α_i ≠ 0$ . The same argument with $[∂_i, D]$ and (2.12) produces a non-zero element of $I$ of degree at most $k-1$ , again a contradiction.

Hence $k = 0$ and $I = A_n$ .

The descent in one step

The same proof can be run without minimality. Take any non-zero $D ∈ I$ , choose a top-degree pair $(α,β)$ with $c_αβ ≠ 0$ , and apply the composite operator in (2.15). Every other monomial of $D$ is annihilated - lower-degree ones because the total order of differentiation exceeds their degree, other top-degree ones because $∂_x^α∂_ξ^β$ kills $x^α'ξ^β'$ unless $α' ≥ α$ and $β' ≥ β$ componentwise, which at equal total degree forces $(α',β') = (α,β)$ . What survives is the non-zero scalar $α!,β!,c_αβ ∈ I$ .

Every endomorphism of $A_n$ is injectiveCoutinho (2.2.2)

Let $φ : A_n → A_n$ be a $K$ -algebra homomorphism with $φ(1) = 1$ . Its kernel is a two-sided ideal not containing $1$ , hence not all of $A_n$ , hence zero. So $φ$ is injective.

Whether every such $φ$ is also surjective is the Dixmier conjecture, open since 1968 for every $n ≥ 1$ . Injectivity is elementary; surjectivity is not known. The conjecture is now known to be equivalent to the Jacobian conjecture in a suitable stable sense.

The centre of $A_n$ is $K$

If $D$ is central then $[x_i,D] = [∂_i,D] = 0$ for all $i$ , so by (2.14) all $2n$ partial derivatives of $P_D$ vanish. In characteristic zero this forces $P_D$ to be constant, so $D ∈ K$ . This is used repeatedly, for instance in the proof of Bernstein's inequality, where an operator commuting with all generators is concluded to be a scalar.

Every non-zero module is faithful

If $M ≠ 0$ is a left $A_n$ -module, its annihilator $Ann(M) = D : DM = 0$ is a two-sided ideal. It cannot be all of $A_n$ , since $1$ acts as the identity on $M ≠ 0$ . So it is zero: no non-zero operator kills all of $M$ .

Key Equations

The two exact bracket formulas that drive the descent, for a canonical monomial:

[∂_i,; x^α ∂^β] = α_i, x^α - e_i∂^β,

(2.12)

[x_i,; x^α ∂^β] = -,β_i, x^α∂^β - e_i.

(2.13)

Extending linearly and writing $P_D$ for the coefficient polynomial of $D$ :

P_[∂_i, D] = (∂ P_D)/(∂ x_i), P_[x_i, D] = -,(∂ P_D)/(∂ ξ_i).

(2.14)

Iterating over a multi-index pair $(α,β)$ of maximal length picks out a single coefficient:

(ad,∂)^α(-ad,x)^β(D) = α!,β!;c_αβ ; ∈ K,

(2.15)

whenever $|α|+|β| = (D)$ , where $ad y$ denotes $D ↦ [y,D]$ and $α! = α_1! ⋅ sα_n!$ .

The characteristic-zero hypothesis enters only through the non-vanishing of that factorial:

α!,β! ≠ 0 in K ⟺ char,K = 0 or char,K > (α_i,β_i).

(2.16)

Variable Definitions

$K$: the ground field, of characteristic zero
$A_n$: the $n$ -th Weyl algebra over $K$
$I$: a two-sided ideal of $A_n$
$D$: an element of $I$ of least degree among the non-zero elements
$c_αβ$: the coefficient of $x^α∂^β$ in the canonical form of $D$
$P_D(x,ξ)$: the commutative polynomial recording all the coefficients of $D$ , obtained by replacing $∂_i$ with $ξ_i$
$ad y$: the adjoint action $D ↦ [y, D] = yD - Dy$
$e_i$: the multi-index with $1$ in slot $i$ and $0$ elsewhere
$Z(A_n)$: the centre of $A_n$ , equal to $K$ in characteristic zero

Properties and Behaviour

Simplicity is used less as a fact about ideals than as a supply of leverage. The most-used consequences:

Consequences of simplicity and where they are used in the collection.
Consequence	Reason	Used for
Every non-zero module is faithful	the annihilator is two-sided	Bernstein's inequality
Every unital endomorphism is injective	the kernel is two-sided	automorphisms, the Dixmier conjecture
The centre is $K$	central elements have vanishing brackets	the faithfulness lemma in dimension theory
No non-zero finite-dimensional module	trace argument plus simplicity	representation theory
$A_n$ is not semisimple	simple artinian would force a matrix ring	structure theory
$A_n$ is a simple domain	combines with additivity of degree	distinguishes $A_n$ from matrix algebras

Simple but not artinian, hence not semisimple

The descending chain $A_1 ⊃ A_1x ⊃ A_1x^2 ⊃ …$ of left ideals is strictly decreasing, so $A_n$ does not satisfy the descending chain condition. By the Artin-Wedderburn theorem a simple artinian ring is a matrix ring over a division ring; $A_n$ is simple but not artinian, and being a domain it is not a matrix ring for $n ≥ 1$ . It is, however, Noetherian: the ascending chain condition does hold.

Simplicity of tensor products and quotients

$A_n ⊗ _K A_m ≅ A_n+m$ is simple, consistent with the general fact that a tensor product of a simple $K$ -algebra with centre $K$ and any simple $K$ -algebra is simple. Since $A_n$ has no proper non-zero two-sided ideals, it has no proper quotient rings other than the zero ring - so there is no useful notion of "reducing $A_n$ modulo an ideal", and every non-trivial construction of $A_n$ -modules must go through one-sided ideals.

Examples and Special Cases

Every non-zero operator generates everything, two-sidedly

For $D = x$ in $A_1$ : $[∂, x] = 1$ , so the two-sided ideal generated by $x$ contains $1$ . For $D = ∂$ : $[x,∂] = -1$ . For $D = x^k$ : $k$ brackets with $∂$ give $k!$ . Each is an instance of (2.15) with the descent taking $(D)$ steps.

One-sided ideals are abundant

The left ideals $A_1∂$ , $A_1 x$ , $A_1(x∂ - λ)$ for $λ ∈ K$ , and $A_1∂^2 + A_1(x∂ - 1)$ are pairwise distinct proper non-zero left ideals. The quotients $A_1/A_1∂ ≅ K[x]$ and $A_1/A_1(x∂-λ)$ are the basic examples of $A_1$ -modules. Simplicity does not restrict this supply in the slightest.

A simple ring that is not a domain

The matrix algebra $M_2(K)$ is simple but has zero divisors and non-trivial idempotents. Conversely $K[x]$ is a domain that is very far from simple. The two conditions are independent, and $A_n$ happens to satisfy both.

A Weyl-like algebra that is not simple

$A_1(Z)$ , the $ℤ$ -subalgebra of $A_1(Q)$ generated by $x$ and $∂$ , is a domain but is not simple: for any prime $p$ , the set $pA_1(Z)$ is a proper non-zero two-sided ideal. The proof above breaks at the last line, where a non-zero constant is declared to be a unit - true over a field, false over $ℤ$ .

Worked Example

Bracketing $(x∂)^2$ down to a constant in $A_1$

Step 1 - fix the operator and put it in canonical form
Let $E = x∂$ be the Euler operator and take $D = E^2$ . Using $∂ x = x∂ + 1$ ,

$D = x∂ x∂ = x(x∂ + 1)∂ = x^2∂^2 + x∂.$

So $(D) = 4$ and the coefficient polynomial is $P_D = x^2ξ^2 + xξ$ . Let $I$ be the two-sided ideal generated by $D$ . We show $I = A_1$ by explicit descent, without appealing to the theorem.
Step 2 - bracket with $∂$ twice
By (2.14), bracketing with $∂$ differentiates $P_D$ in $x$ . Directly:

$[∂, D] = [∂, x^2]∂^2 + [∂, x]∂ = 2x∂^2 + ∂,$

$[∂, 2x∂^2 + ∂] = 2∂^2.$

Check against the polynomial picture: $∂_x(x^2ξ^2 + xξ) = 2xξ^2 + ξ$ and $∂_x^2(x^2ξ^2+xξ) = 2ξ^2$ . The two agree. Degrees have fallen $4 → 3 → 2$ , one per bracket, and nothing has become zero.
Step 3 - bracket with $x$ twice
Now (2.13) applies, with the minus sign:

$[x, 2∂^2] = -2,[∂^2, x] = -4∂, [x, -4∂] = -4,[x,∂] = -4 ⋅ (-1) = 4.$

Here $[∂^2,x] = 2∂$ and $[x,∂] = -[∂,x] = -1$ . The polynomial picture agrees: applying $-∂_ξ$ twice to $2ξ^2$ gives $4$ . The degrees have fallen $2 → 1 → 0$ .
Step 4 - conclude
The chain $D → [∂,D] → [∂,[∂,D]] → … → 4$ stays inside any two-sided ideal containing $D$ . So $4 ∈ I$ , and since $4$ is a unit in $ℚ$ , $I = A_1$ . Formula (2.15) predicts the value in advance: the top pair is $(α,β) = (2,2)$ with $c_22 = 1$ , giving $α!β!c_αβ = 2! ⋅ 2! ⋅ 1 = 4$ . It matches.
Step 5 - the same descent over $Z_2$ stalls
Now read the identical computation over $Z_2$ , in the algebra presented by generators and relations so that the canonical monomials remain a basis. The predicted constant is $2! ⋅ 2! = 4 ≡ 0$ , and indeed $[x, 2∂^2] = -4∂ = 0$ : the descent dies one step early and this route produces nothing.

That is not yet a proof of non-simplicity, but the obstruction is real. Over $Z_p$ the element $x^p$ satisfies $[∂, x^p] = px^p-1 = 0$ and $[x, x^p] = 0$ , so $x^p$ is central and the two-sided ideal it generates is proper - it consists of operators of degree at least $p$ , so it does not contain $1$ . Simplicity fails outright. See the positive characteristic page.

Result

In $A_1(Q)$ , four brackets take $D = (x∂)^2 = x^2∂^2 + x∂$ to the constant $4$ , so the two-sided ideal generated by $D$ is all of $A_1$ . The constant is exactly $α!,β!,c_αβ = 4$ as (2.15) predicts. Over $Z_2$ the same constant is $0$ and the descent fails; over $Z_p$ the algebra is genuinely not simple, since $x^p$ is central.

Applications and Industry Use

In a mathematics topic, this section covers downstream use inside mathematics, computing and engineering rather than a manufactured product.

Simplicity is invoked at the following points in this collection:

In the proof of Bernstein's inequality, where the key lemma concludes that an operator commuting with every generator is a scalar, and that a scalar killing a non-zero module is zero.
In the simplicity of P0 as an P1-module, a different but analogous descent, this time on the degree of a polynomial rather than of an operator.
In showing that no non-zero finite-dimensional representation exists, since a non-zero representation would be faithful and its kernel argument would force a contradiction with the trace of $[∂,x] = 1$ .
In the study of automorphisms and endomorphisms, via Corollary (2.2.2).
In holonomic module theory, where faithfulness of every non-zero module is what makes dimension a meaningful invariant.

Outside the collection, the same phenomenon - a simple infinite-dimensional algebra of differential operators - is the algebraic backbone of the theory of primitive ideals in enveloping algebras, where quotients $U(g)/J$ by primitive ideals often turn out to be closely related to Weyl algebras.

Limits of Validity

Three hypotheses are load-bearing.

$K$ must be a field. Over a commutative ring $R$ that is not a field, $A_n(R)$ has the proper two-sided ideal $mA_n(R)$ for every proper ideal $𝔪$ of $R$ . The descent still reaches a non-zero constant; that constant just is not invertible.
$K$ must have characteristic zero. For $char,K = p$ the elements $x_i^p$ and $∂_i^p$ are central in the generators-and-relations version of $A_n$ , and the two-sided ideal generated by $x_1^p$ is proper. The algebra is then a finitely generated module over its centre, of rank $p^2n$ - about as far from simple as a domain can be.
Simplicity says nothing about one-sided ideals. It is not a finiteness statement. $A_n$ has infinitely many left ideals, and classifying even the cyclic modules $A_1/A_1P$ is a serious problem.

What generalises

For a smooth irreducible affine variety $X$ over a field of characteristic zero, $D(X)$ is simple. The proof is not the degree descent - there is no Bernstein filtration on a general $X$ - but a localisation argument using that $X$ is covered by open sets on which $𝒟$ looks like a Weyl algebra. On singular varieties simplicity can fail: $D(X)$ for the cuspidal cubic $y^2 = x^3$ is not simple, and this is one of the standard warnings against extending affine-space intuition. See differential operators on an affine variety.

Failure Modes and Common Mistakes

Forgetting to check that the bracket is non-zero

The descent needs $[x_i, D] ≠ 0$ , not merely $[x_i,D] < D$ - the zero operator has smaller degree too, and would give no contradiction. Establishing non-vanishing is where the coefficient $β_i$ must be invertible in $K$ , and it is the only place characteristic zero is used. A proof that skips this step is not a proof; it is the characteristic- $p$ statement, which is false.

Reading "simple" as "every ideal is trivial"

Only two-sided ideals are trivial. Confusing this with the one-sided case would make $A_n$ a division ring, which it is not: the units are exactly $K^ ×$ . Whenever a source says "ideal" in noncommutative ring theory, check which side is meant.

Assuming simple implies semisimple or artinian

For finite-dimensional algebras the two coincide and both mean "matrix ring over a division ring". $A_n$ is infinite dimensional, simple, Noetherian, and not artinian. Any structural argument that quotes Artin-Wedderburn for $A_n$ is invalid, and this mistake is the root of the false expectation that $A_n$ should have finite-dimensional representations.

Concluding from injectivity of endomorphisms that they are automorphisms

Injectivity is Corollary (2.2.2) and is elementary. Surjectivity is the Dixmier conjecture and is open for every $n ≥ 1$ . For finite-dimensional algebras injective implies surjective by dimension count; $A_n$ is infinite dimensional and that argument is unavailable. Do not upgrade the corollary.

Historical Notes

That the canonical commutation relation forces a simple algebra was implicit in the uniqueness theorems of early quantum mechanics: Stone and von Neumann's 1930-31 theorem on the uniqueness of irreducible representations of the Weyl relations is the analytic counterpart of algebraic simplicity. The purely algebraic statement, with the degree descent as proof, is standard by the 1960s.

Dixmier's 1968 paper on $A_1$ is the reference that made the ideal-theoretic facts systematic: it records simplicity, the classification of units, the automorphism group of $A_1$ , and poses the question of whether injective endomorphisms are surjective. That question, now the Dixmier conjecture, is still open, and Tsuchimoto's and Belov-Kanel and Kontsevich's work in the 2000s showed it is equivalent to the Jacobian conjecture in a stable form.

The positive-characteristic picture was clarified in the 1970s and 1980s, notably in work of S. P. Smith and others, and it is now a substantial subject in its own right: the Weyl algebra over $Z_p$ is an Azumaya algebra over its centre away from a small locus, which is the technical entry point to the theory of $D$ -modules in characteristic $p$ .

Key Takeaways

Key takeaways

$A_n(K)$ with $char,K = 0$ has exactly two two-sided ideals: $0$ and $A_n$ .
The proof is a descent: brackets with the generators lower degree by one, and a two-sided ideal is closed under them.
On coefficient polynomials, $ad,∂_i$ is $∂/∂ x_i$ and $ad,x_i$ is $-∂/∂ξ_i$ ; these identities are exact, not just to leading order.
Characteristic zero is used at exactly one point: to know that the factorials $α! β!$ appearing in (2.15) are non-zero.
Consequences: the centre is $K$ ; every unital endomorphism is injective; every non-zero module is faithful.
Simplicity constrains only two-sided ideals. $A_n$ has an abundance of left ideals and is not a division ring, not artinian and not semisimple.
Injectivity of endomorphisms does not give surjectivity: that is the Dixmier conjecture, still open.

FAQs

Why does simplicity not make

A_n

a field or a division ring?

Because that implication needs commutativity. A commutative simple ring is a field, since for $a ≠ 0$ the ideal $(a)$ is everything, so $a$ has an inverse. In the noncommutative case the two-sided ideal generated by $a$ is $∑ b_i a c_i$ , and getting $1$ out of that says nothing about $a$ having an inverse. For $a = x$ in $A_1$ we have $∂ x - x∂ = 1$ , which certainly does not make $x$ invertible.

Where does the proof use that

K

has characteristic zero?

Exactly once, in the assertion that $β_i c_αβ ≠ 0$ when $β_i ≠ 0$ as an integer. In characteristic $p$ the integer $β_i$ can be zero in $K$ , the bracket vanishes, and the descent stops. This is not a defect of the proof: the theorem itself is false in characteristic $p$ .

Is the descent effective - can I actually compute the constant?

Yes, and formula (2.15) gives it in closed form: pick any top-degree pair $(α,β)$ with $c_αβ ≠ 0$ , apply $ad,∂$ and $-ad,x$ the corresponding number of times, and the answer is $α!,β!,c_αβ$ . The worked example carries this out and the predicted value $4$ is confirmed by direct computation.

Does simplicity imply that

A_n

has a unique simple module?

No, and this is a common confusion. A simple ring has trivial two-sided ideal structure; a simple module is one with no proper non-zero submodules. Simple artinian rings do have a unique simple module up to isomorphism, but $A_n$ is not artinian and has a rich supply of non-isomorphic simple modules - for instance $K[x]$ and the twisted modules $A_1/A_1(x∂-λ)$ for suitable $λ$ .

If every non-zero module is faithful, why is that useful?

Because it turns a statement about the size of the algebra into a statement about the size of the module. The whole of $A_n$ must act by distinct operators on $M$ , and $_K$ of the degree- $m$ part of $A_n$ grows like $m^2n$ . Comparing that growth to the growth of $M$ is precisely the proof of Bernstein's inequality.

What is the two-sided ideal structure in characteristic

p

Rich. The centre of the generators-and-relations Weyl algebra over $Z_p$ is the polynomial ring $Z_p[x_1^p, … ,x_n^p,∂_1^p, … ,∂_n^p]$ , and the algebra is a free module of rank $p^2n$ over it. Every ideal of the centre generates a two-sided ideal, so there are as many as in a polynomial ring in $2n$ variables. Simplicity fails in the strongest possible way.

Is there a version of simplicity for the ring

B_n(K)

of operators with rational coefficients?

Yes. $B_n(K)$ , the $K(x_1, … ,x_n)$ -algebra generated by the $∂_i$ , is also a simple domain in characteristic zero, and Coutinho develops this in the exercises to Ch. 2 using the order rather than the degree. For $n = 1$ it has the additional feature of a left division algorithm, so every left ideal is principal - a property $A_1$ does not have.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 2 §2, Theorem (2.2.1) and Corollary (2.2.2).
J. Dixmier, Sur les algèbres de Weyl, Bulletin de la Société Mathématique de France 96 (1968), 209-242 - simplicity, units, automorphisms of $A_1$ and the origin of the Dixmier conjecture.
J.-E. Björk, Rings of Differential Operators, North-Holland Mathematical Library 21, 1979 - Ch. 1, the ideal theory of $A_n$ and of rings of differential operators.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, revised edition, American Mathematical Society, 2001 - Ch. 1 and Ch. 8, simple Noetherian rings and the Weyl algebra.
J. T. Stafford, Module structure of Weyl algebras, Journal of the London Mathematical Society 18 (1978), 429-442 - two-generation of left ideals, a sharp contrast with the two-sided picture.
S. P. Smith, Differential operators on commutative algebras, in Ring Theory (Antwerp 1985), Lecture Notes in Mathematics 1197, Springer, 1986 - the positive characteristic case.
Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka Journal of Mathematics 42 (2005), 435-452 - the link between the Dixmier and Jacobian conjectures via characteristic $p$ .
ISO 80000-2:2019, Quantities and units - Part 2: Mathematics - notation for commutators, multi-indices and factorials.

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