Simplicity and Commutative Bases of Derivations in Polynomial and Power Series Rings

Abstract

The first part of the paper will describe a recent result of Retert in (2006) for 𝑘[𝑥₁,…,𝑥_𝑛] and 𝑘[[𝑥₁,…,𝑥_𝑛]]. This result states that if 𝔇 is a set of commute 𝑘-derivations of 𝑘[𝑥,𝑦] such that both 𝜕_𝑥∈𝔇 and the ring is 𝔇-simple, then there is 𝑑∈𝔇 such that 𝑘[𝑥,𝑦] is {𝜕_𝑥,𝑑}-simple. As for applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations.

1. Introduction

Let 𝑘 be a field of characteristic zero and 𝑅 denote either the ring 𝑘[𝑥₁,…,𝑥_𝑛] of polynomials over 𝑘 or the ring 𝑘[[𝑥₁,…,𝑥_𝑛]] of formal power series over 𝑘.

A 𝑘-derivation 𝑑∶𝑅→𝑅 of 𝑅 is a 𝑘-linear map such that 𝑑(𝑎𝑏)=𝑑(𝑎)𝑏+𝑎𝑑(𝑏) for any 𝑎,𝑏∈𝑅. Denoting by Der_𝑘(𝑅) the set of all 𝑘-derivations of 𝑅, let 𝔇⊂Der_𝑘(𝑅) be a nonempty family of 𝑘-derivations. An ideal 𝐼 of 𝑅 is called 𝔇-stable if 𝑑(𝐼)⊂𝐼 for all 𝑑∈𝔇. For example, the ideals 0 and 𝑅 are always 𝔇-stable. If 𝑅 has no other 𝔇-stable ideal it is called 𝔇-simple. When 𝔇={𝑑}, 𝑑 is often called a simple derivation.

The commuting derivations have been studied by several authors: Li and Du [1], Maubach [2], Nowicki [3], Petravchuk [4], Retert [5], Van den Essen [6]. For example, it is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector; in [4], Petravchuk proved an analogous statement for derivation of 𝑘[𝑥,𝑦] over any field 𝑘 of characteristic zero. More explicitly, if two derivations of 𝑘[𝑥,𝑦] are linearly independent over 𝑘 and commute, then they have a common Darboux polynomial or they are Jacobian derivations; in [1], the authors proved the same result for 𝑘[𝑥₁,…,𝑥_𝑛] and 𝑘[[𝑥₁,…,𝑥_𝑛]]. However, we observe that this result has already been proved by Nowicki in (Now86) for both rings. Another interesting result was proved by Nowicki in [3, Theorem  5] which says that the famous Jacobian conjecture in 𝑘[𝑥₁,…,𝑥_𝑛] is equivalent to the assertion that every commutative basis of Der_𝑘(𝑅) is locally nilpotent.

Let 𝔇 be a set of commute 𝑘-derivations of 𝑘[𝑥]; then 𝑘[𝑥] is 𝔇-simple if and only if it is 𝑑-simple for some 𝑘-derivation 𝑑∈𝔇 (see [5, Corollary  2.10]). In 𝑘[𝑥,𝑦], as pointed out in [5], up to scalar multiples, these are only sets 𝔇 of two commuting, nonsimple 𝑘-derivations such that both 𝜕_𝑥∈𝔇 and 𝑅 is 𝔇-simple. Motivated by this, we analyze this result in [5] for 𝑅 and then we propose some connections with known results on commutative bases of derivations in 𝑅. More precisely, the derivations 𝜕_𝑥1,…,𝜕_𝑥𝑛−1 are not simple 𝑘-derivations of 𝑅; however, as will be shown, they can be part of a set 𝔇 of 𝑛 commuting, nonsimple 𝑘-derivations such that 𝑅 is 𝔇-simple. A trivial example is 𝔇={𝜕_𝑥1,…,𝜕_𝑥𝑛}. Using the notations in [3], we give a nontrivial commutative base containing only nonsimple 𝑘-derivations of the free 𝑅-module Der_𝑘(𝑅) such that 𝑅 is 𝔇-simple and if the Jacobian conjecture is true in 𝑘[𝑥₁,…,𝑥_𝑛], as a consequence of [3, Theorem  5], we obtain a family of locally nilpotent derivations.

2. Commuting Derivations and Simplicity

Lemma 1. The set of all 𝑘-derivations of 𝑅 that commute with 𝜕_𝑥1,…,𝜕_𝑥𝑛−1 is 𝑝𝑥_𝑛𝜕_𝑥1+𝑞₁𝑥_𝑛𝜕_𝑥2+⋯+𝑞_𝑛−1𝑥_𝑛𝜕_𝑥𝑛 (1) such that 𝑝,  𝑞₁,…,𝑞_𝑛−1∈𝑘[𝑥_𝑛] (or 𝑘[[𝑥_𝑛]] if 𝑅=𝑘[[𝑥₁,…,𝑥_𝑛]]).

Proof. It is clear that all derivations of this form commute with the derivations 𝜕_𝑥1,…,𝜕_𝑥𝑛−1. For the converse, let 𝑑^∗=𝑝^∗𝑥₁,…,𝑥_𝑛𝜕_𝑥1+𝑞^∗₁𝑥₁,…,𝑥_𝑛𝜕_𝑥2+⋯+𝑞^∗_𝑛−1𝑥₁,…,𝑥_𝑛𝜕_𝑥𝑛 (2) be a 𝑘-derivation of 𝑅 that commutes with 𝜕_𝑥1,…,𝜕_𝑥𝑛−1. Then 0=𝑑^∗𝜕_𝑥𝑖𝑥₁=𝜕_𝑥𝑖𝑑^∗𝑥₁=𝜕_𝑥𝑖𝑝^∗𝑥₁,…,𝑥_𝑛 (3) for all 𝑖=1,…,𝑛−1. Thus, 𝑝^∗(𝑥₁,…,𝑥_𝑛)∈𝑘[𝑥_𝑛](𝑘[[𝑥_𝑛]]). Similarly, we can prove that 𝑞^∗_𝑖(𝑥₁,…,𝑥_𝑛)∈𝑘[𝑥_𝑛](𝑘[[𝑥_𝑛]]).

Let 𝔇={𝛿₁,…,𝛿_𝑠} be any finite set of 𝑘-derivation of 𝑅 that commutes with 𝜕_𝑥1,…,𝜕_𝑥𝑛−1 but not necessarily with each other. By Lemma 1, each 𝛿_𝑖 is of the form 𝛿_𝑖=𝑝_𝑖𝑥_𝑛𝜕_𝑥1+𝑞_(1,𝑖)𝑥_𝑛𝜕_𝑥2+⋯+𝑞_{(𝑛−1,𝑖)}𝑥_𝑛𝜕_𝑥𝑛. (4)

We denote by 𝑣_𝑛−1(𝑥_𝑛) the greatest common divisor of 𝑞_(𝑛−1,1),…,𝑞_{(𝑛−1,𝑠)}. We have the following characterization for the simplicity of 𝑅.

Lemma 2. Using the notations above, 𝑅 is 𝔇-simple if and only if 𝑣_𝑛−1(𝑥_𝑛) is a unit in 𝑘[𝑥_𝑛] (or 𝑘[[𝑥_𝑛]]).

Proof. If all 𝑞_{(𝑛−1,𝑖)}=0, so all the 𝑘-derivations in 𝔇 stabilize the nonzero ideal ⟨𝑥_𝑛⟩; in this case, 𝑅 is not 𝔇-simple. Then we assume that some 𝑞_{(𝑛−1,𝑖)}≠0. If 𝑣_𝑛−1(𝑥_𝑛) is not a unit, each 𝛿_𝑖 stabilizes the nontrivial ideal ⟨𝑣_𝑛−1(𝑥_𝑛)⟩. Therefore 𝑅 is not 𝔇-simple.
Conversely, assume that 𝑣_𝑛−1(𝑥_𝑛) is a unit and notice that, in this case, there are polynomials 𝑟_𝑖(𝑥_𝑛) such that ^𝑠_𝑖=1𝑟_𝑖𝑥_𝑛𝑞_{(𝑛−1,𝑖)}𝑥_𝑛=𝑣_𝑛−1𝑥_𝑛, (5) multiplicand by the inverse of 𝑣_𝑛−1(𝑥_𝑛), we may assume that ∑^𝑠_𝑖=1𝑟_𝑖(𝑥_𝑛)𝑞_{(𝑛−1,𝑖)}(𝑥_𝑛)=1. Without loss of generality, let 𝛿₁=𝜕_𝑥1,…,𝛿_𝑛−1=𝜕_𝑥𝑛−1; thus, ^𝑠_𝑖=𝑛𝑟_𝑖𝑥_𝑛𝑞_{(𝑛−1,𝑖)}𝑥_𝑛=1. (6) Let 𝐼 be a 𝔇-ideal. Then since 𝐼 is stabilized by each 𝛿_𝑖, 𝐼 is stabilized by 𝑟_𝑖(𝑥_𝑛)𝛿_𝑖 and, then, by the 𝑘-derivation ^𝑠_𝑖=𝑛𝑟_𝑖𝑥_𝑛𝛿_𝑖=^𝑠_𝑖=𝑛𝑟_𝑖𝑥_𝑛𝑝_𝑖𝑥_𝑛𝜕_𝑥1+⋯+^𝑠_𝑖=𝑛𝑟_𝑖𝑥_𝑛𝑞_{(𝑛−1,𝑖)}𝑥_𝑛𝜕_𝑥𝑛. (7) Therefore, 𝐼 is stabilized by 𝜕_𝑥1,…,𝜕_𝑥𝑛−1 and a 𝑘-derivation of the form 𝑢₁𝑥_𝑛𝜕_𝑥1+⋯+𝑢_𝑛−1𝑥_𝑛𝜕_𝑥𝑛−1+𝜕_𝑥𝑛 (8) for 𝑢_𝑖(𝑥_𝑛)∈𝑘[𝑥_𝑛](𝑘[[𝑥_𝑛]]). Thus 𝐼 is stabilized by 𝜕_𝑥1,…,𝜕_𝑥𝑛−1,𝜕_𝑥𝑛 and, then, we deduce that 𝐼 must be a trivial ideal.

Note that until now we only assume that all the 𝑘-derivations commute with 𝜕_𝑥1,…,𝜕_𝑥𝑛−1 not that all elements commute with each other. Using the previous lemmas, the following theorem will show that if 𝑅 is 𝔇-simple under a set 𝔇 of commuting 𝑘-derivations that contains 𝜕_𝑥1,…,𝜕_𝑥𝑛−1, then 𝑅 is simple under a subset of 𝑛 commuting nonsimple 𝑘-derivations.

Theorem 3. Let 𝔇 be a set of 𝑘-derivations of 𝑅 such that 𝜕_𝑥1,…,𝜕_𝑥𝑛−1∈𝔇. Then the derivations of 𝔇 commute with each other if and only if one of the following two cases holds.(a)Each element 𝛿_𝑖 of 𝔇 has the form 𝛿_𝑖=ℎ^𝑖₁(𝑥_𝑛)𝜕_𝑥1+⋯+ℎ^𝑖_𝑛−1(𝑥_𝑛)𝜕_𝑥𝑛−1, for some ℎ^𝑖_𝑗(𝑥_𝑛)∈𝑘[𝑥_𝑛] (or 𝑘[[𝑥_𝑛]]).(b)There exists 𝑣₁(𝑥_𝑛),…,𝑣_𝑛(𝑥_𝑛)∈𝑘[𝑥_𝑛] (or 𝑘[[𝑥_𝑛]]), such that for each 𝛿_𝑖∈𝔇 there are scalars 𝜆_𝑖,𝑐^𝑖₁,…,𝑐^𝑖_𝑛−1∈𝑘 such that 𝛿_𝑖=^𝑛−1_𝑙=1𝜆_𝑖𝑣_𝑙𝑥_𝑛+𝑐^𝑖_𝑙𝜕_𝑥𝑙+𝜆_𝑖𝑣_𝑛𝑥_𝑛𝜕_𝑥𝑛. (9)

If, in addition, 𝑅 is 𝔇-simple, then 𝑣_𝑛(𝑥_𝑛) must be some nonzero scalar 𝛽 and, also, some 𝜆_𝑗 is not zero. In this case, 𝑅 is also simple under the subset 𝜕_𝑥1,…,𝜕_𝑥𝑛−1,^𝑛−1_𝑙=1𝜆_𝑗𝑣_𝑙𝑥_𝑛+𝑐^𝑗_𝑙𝜕_𝑥𝑙+𝜆_𝑗𝛽𝜕_𝑥𝑛. (10)

Proof. If either of the conditions is met, it is clear that all 𝑘-derivations of 𝔇 will commute.
Conversely, let 𝔇={𝛿₁,…,𝛿_𝑚} be according to the hypotheses of the theorem. By Lemma 1, each 𝛿_𝑖 is of the form 𝛿_𝑖=𝑝_𝑖(𝑥_𝑛)𝜕_𝑥1+𝑞_(1,𝑖)(𝑥_𝑛)𝜕_𝑥2+⋯+𝑞_{(𝑛−1,𝑖)}(𝑥_𝑛)𝜕_𝑥𝑛.
If all 𝑞_{(𝑛−1,𝑖)}(𝑥_𝑛) are zero, then the first case holds. So, without loss of generality, we assume that 𝑞_(𝑛−1,1)(𝑥_𝑛)≠0 and observe that 𝛿_𝑖𝛿₁𝑥_𝑛=𝛿_𝑖𝑞_(𝑛−1,1)𝑥_𝑛=𝑞_{(𝑛−1,𝑖)}𝑥_𝑛𝜕_𝑥𝑛𝑞_(𝑛−1,1)𝑥_𝑛,𝛿₁𝛿_𝑖𝑥_𝑛=𝛿₁𝑞_{(𝑛−1,𝑖)}𝑥_𝑛=𝑞_(𝑛−1,1)𝑥_𝑛𝜕_𝑥𝑛𝑞_{(𝑛−1,𝑖)}𝑥_𝑛. (11)
Since 𝛿_𝑖 and 𝛿₁ commute, 𝑞_{(𝑛−1,𝑖)}𝑥_𝑛𝜕_𝑥𝑛𝑞_(𝑛−1,1)𝑥_𝑛=𝑞_(𝑛−1,1)𝑥_𝑛𝜕_𝑥𝑛𝑞_{(𝑛−1,𝑖)}𝑥_𝑛. (12) Then, this equation must also hold in the ring of fractions; hence we deduce that 𝑞_{(𝑛−1,𝑖)}𝑥_𝑛𝜕_𝑥𝑛𝑞_(𝑛−1,1)𝑥_𝑛−𝑞_(𝑛−1,1)𝑥_𝑛𝜕_𝑥𝑛𝑞_{(𝑛−1,𝑖)}𝑥_𝑛𝑞_(𝑛−1,1)𝑥_𝑛²=0. (13) In other words, 𝑞_{(𝑛−1,𝑖)}𝑥_𝑛𝑞_(𝑛−1,1)𝑥_𝑛^′=0. (14)
Then there is some 𝜆_𝑖∈𝑘 such that 𝑞_{(𝑛−1,𝑖)}(𝑥_𝑛)=𝜆_𝑖𝑞_(𝑛−1,1)(𝑥_𝑛).
Now, we observe that 𝛿_𝑖𝛿₁𝑥₁=𝛿_𝑖𝑝₁𝑥_𝑛=𝑞_{(𝑛−1,𝑖)}𝑥_𝑛𝜕_𝑥𝑛𝑝₁𝑥_𝑛,𝛿₁𝛿_𝑖𝑥₁=𝛿₁𝑝_𝑖𝑥_𝑛=𝑞_(𝑛−1,1)𝑥_𝑛𝜕_𝑥𝑛𝑝_𝑖𝑥_𝑛. (15)
Since 𝛿_𝑖 and 𝛿₁ commute and both 𝑘[𝑥_𝑛] and 𝑘[[𝑥_𝑛]] are domains, 𝜆_𝑖𝜕_𝑥𝑛𝑝₁𝑥_𝑛=𝜕_𝑥𝑛𝑝_𝑖𝑥_𝑛. (16)
Then there is some 𝑐^𝑖₁∈𝑘 such that 𝑝_𝑖(𝑥_𝑛)=𝜆_𝑖𝑝₁(𝑥_𝑛)+𝑐^𝑖₁. Finally, making the same argument for the other variables, we prove the desired result.
Now we suppose, in addition, that 𝑅 is 𝔇-simple. By Lemma 2, the greatest common divisor of 𝑞_(𝑛−1,1)(𝑥_𝑛),…,𝑞_{(𝑛−1,𝑚)}(𝑥_𝑛) must be a unit. However, we have demonstrated that all the 𝑞_{(𝑛−1,𝑖)}(𝑥_𝑛) are scalar multiples; then at least one of the 𝑞_{(𝑛−1,𝑖)}(𝑥_𝑛) must be a unit; we assume that 𝑞_{(𝑛−1,𝑗)}(𝑥_𝑛) is a unit. Since 𝐼 is stabilized by 𝜕_𝑥1,…,𝜕_𝑥𝑛−1,𝑝_𝑗𝑥_𝑛𝜕_𝑥1+𝑞_(1,𝑗)𝑥_𝑛𝜕_𝑥2+⋯+𝑞_{(𝑛−1,𝑗)}𝑥_𝑛𝜕_𝑥𝑛, (17)𝐼 must be trivial because, in this case, 𝐼 is stabilized by 𝜕_𝑥1,…,𝜕_𝑥𝑛−1,𝜕_𝑥𝑛. Therefore, 𝑅 is {𝜕_𝑥1,…,𝜕_𝑥𝑛−1,𝑝_𝑗(𝑥_𝑛)𝜕_𝑥1+𝑞_(1,𝑗)(𝑥_𝑛)𝜕_𝑥2+⋯+𝑞_{(𝑛−1,𝑗)}(𝑥_𝑛)𝜕_𝑥𝑛}-simple which completes the proof.

Remark 4. Nowicki in [7, Theorem  2.5.5] proved that every 𝑘-derivation of a commutative base of Der_𝑘(𝑘[𝑥₁,…,𝑥_𝑛]) is a special 𝑘-derivation. This means that the divergence 𝑑^∗ of 𝑑 is 0. Moreover, it is easy to prove that the set 𝜕_𝑥1,…,𝜕_𝑥𝑛−1,^𝑛−1_𝑙=1𝜆_𝑗𝑣_𝑙𝑥_𝑛+𝑐^𝑗_𝑙𝜕_𝑥𝑙+𝜆_𝑗𝛽𝜕_𝑥𝑛 (18) obtained by the previous theorem is a commutative base of Der_𝑘(𝑘[𝑥₁,…,𝑥_𝑛]). Thus, in particular, ∑^𝑛−1_𝑙=1(𝜆_𝑗𝑣_𝑙(𝑥_𝑛)+𝑐^𝑗_𝑙)𝜕_𝑥𝑙+𝜆_𝑗𝛽𝜕_𝑥𝑛 is a special derivation. However, this is easily verified since 𝑑^∗=^𝑛−1_𝑙=1𝜕_𝑥𝑙𝜆_𝑗𝑣_𝑙𝑥_𝑛+𝑐^𝑗_𝑙+𝜕_𝑥𝑛𝜆_𝑗𝛽=0. (19)

Corollary 5. Let 𝔇 be a set of commute 𝑘-derivations of 𝑅 such that 𝑅 is 𝔇-simple and 𝜕_𝑥1,…,𝜕_𝑥𝑛−1∈𝔇. Then, there exists 𝑑∈𝔇 and there exist elements 𝑓₁,…,𝑓_𝑛∈𝑅 such that 𝑑(𝑓_𝑛)=1 and 𝑑(𝑓_𝑖)=0, for any 𝑖=1,…,𝑛−1, so that 𝑅 is {𝜕_𝑥1,…,𝜕_𝑥𝑛−1,𝑑}-simple.

Proof. By the previous theorem, we know that there is 𝑑∈𝔇 of the form 𝑑=^𝑛−1_𝑙=1𝜆_𝑗𝑣_𝑙𝑥_𝑛+𝑐^𝑗_𝑙𝜕_𝑥𝑙+𝜆_𝑗𝛽𝜕_𝑥𝑛 (20) such that 𝑅 is {𝜕_𝑥1,…,𝜕_𝑥𝑛−1,𝑑}-simple. Since 𝛽 and 𝜆_𝑗, in the theorem, are nonzero scalars, we denote 𝑓_𝑛=(𝜆_𝑗𝛽)⁻¹𝑥_𝑛; thus 𝑑(𝑓_𝑛)=1.
Let 𝑓^∗_𝑙∈𝑘[𝑥_𝑛] (or 𝑘[[𝑥_𝑛]]) such that 𝑑(𝑓^∗_𝑙)=𝜆_𝑗𝑣_𝑙(𝑥_𝑛)+𝑐^𝑗_𝑙. Since 𝑐(𝑘)=0, 𝑓^∗_𝑙 exist. Then, let 𝑓_𝑙=𝑥_𝑙−𝑓^∗_𝑙; hence 𝑑(𝑓_𝑙)=0, for any 𝑙=1,…,𝑛−1. This completes the proof.

Remark 6. The previous corollary is a particular case of an important theorem about the characterization of commutative basis of Der_𝑘(𝑅) However, in our case the proof is more evident (see [3, Theorem  2] (Now86)).

For the remainder of this note we assume that 𝑅 is the ring 𝑘[𝑥₁,…,𝑥_𝑛] of polynomials over 𝑘 and {𝜕_𝑥1,…,𝜕_𝑥𝑛−1,𝑑} is as in the previous theorem and also we recall the following definitions.

We recall from [7] that a 𝑘-derivation 𝑑 of 𝑘[𝑥₁,…,𝑥_𝑛] is called locally nilpotent if for each 𝑓∈𝑅 exists a natural number 𝑛 such that 𝑑^𝑛(𝑓)=0 and we say that a basis {𝑑₁,…,𝑑_𝑛} of Der_𝑘(𝑘[𝑥₁,…,𝑥_𝑛]) is locally nilpotent if every derivation 𝑑_𝑖 is locally nilpotent for 𝑖=1,…,𝑛.

We remember also that the Jacobian conjecture states that if 𝐹=(𝐹₁,…,𝐹_𝑛) is a polynomial map such that the Jacobian matrix is invertible, then 𝐹 has a polynomial inverse (see [7]).

Theorem 7 (see [3, Theorem  5]). Let 𝑅=𝑘[𝑥₁,…,𝑥_𝑛] be the polynomial ring in 𝑛 variables over 𝑘. The following conditions are equivalent.(1)The Jacobian conjecture is true in the 𝑛-variable case.(2)Every commutative basis of the 𝑅-module 𝐷𝑒𝑟_𝑘(𝑅) is locally nilpotent.(3)Every commutative basis of the 𝑅-module 𝐷𝑒𝑟_𝑘(𝑅) is locally finite.

Corollary 8. Let 𝔇 be a set of commute 𝑘-derivations of 𝑅 such that 𝑅 is 𝔇-simple, 𝜕_𝑥1,…,𝜕_𝑥𝑛−1∈𝔇, and the Jacobian conjecture is true in 𝑅. Then, there exists 𝑑∈𝔇 such that {𝜕_𝑥1,…,𝜕_𝑥𝑛−1,𝑑} is a locally nilpotent commutative base of the 𝑅-module 𝐷𝑒𝑟_𝑘(𝑅). In particular, 𝑑 is a 𝑘-derivation locally nilpotent.

Proof. The proof is immediate consequence of [3, Theorem  5].

Question. A ring is called 𝑤-differentially simple if it is a simple relative to a family with 𝑤 derivations. Recall that we are assuming 𝑅=𝑘[𝑥₁,…,𝑥_𝑛]; then we know that 𝑅 is 1-differentially simple and dim(𝑅)-differentially simple as well. However, 𝑛=dim(𝑅) is not necessarily the smallest 𝑤 for which such a ring can be 𝑤-differentially simple (see [8]). Thus, one may ask the following: what is the smallest positive integer 𝑤≠1 such that 𝑅 is 𝑤-differentially simple and all 𝑤 derivations are nonsimple and commute?

Acknowledgment

The research of Rene Baltazar was partially supported by CAPES of Brazil.