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

Authors: Rene Baltazar
Journal: ISRN Algebra
 (2013)

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## Abstract

The first part of the paper will describe a recent result of Retert in (2006) forkx1,…,xn and kx1,…,xn. This result states that if 𝔇 is a set of commute k-derivations of kx,y such that both ∂x∈𝔇 and the ring is 𝔇-simple, then there is d∈𝔇 such that kx,y is ∂x,d-simple. As for applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations.


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## Body

## 1. Introduction

Letk be a field of characteristic zero and R denote either the ring k[x1,…,xn] of polynomials over k or the ring k[[x1,…,xn]] of formal power series over k.Ak-derivationd: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 Derk(R) the set of all k-derivations of R, let 𝔇⊂Derk(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,…,xn] and k[[x1,…,xn]]. 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,…,xn] is equivalent to the assertion that every commutative basis of Derk(R) is locally nilpotent.Let𝔇 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 Derk(R) such that R is 𝔇-simple and if the Jacobian conjecture is true in k[x1,…,xn], 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 allk-derivations of R that commute with ∂x1,…,∂xn-1 is
(1){p(xn)∂x1+q1(xn)∂x2+⋯+qn-1(xn)∂xn}
such that p,  q1,…,qn-1∈k[xn] (or k[[xn]] if R=k[[x1,…,xn]]).Proof.
It is clear that all derivations of this form commute with the derivations∂x1,…,∂xn-1. For the converse, let
(2)d*=p*(x1,…,xn)∂x1+q1*(x1,…,xn)∂x2+⋯+qn-1*(x1,…,xn)∂xn
be a k-derivation of R that commutes with ∂x1,…,∂xn-1. Then
(3)0=d*(∂xi(x1))=∂xi(d*(x1))=∂xi(p*(x1,…,xn))
for all i=1,…,n-1. Thus, p*(x1,…,xn)∈k[xn](k[[xn]]). Similarly, we can prove that qi*(x1,…,xn)∈k[xn](k[[xn]]).Let𝔇={δ1,…,δs} 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 δi is of the form
(4)δi=pi(xn)∂x1+q(1,i)(xn)∂x2+⋯+q(n-1,i)(xn)∂xn.We denote byvn-1(xn) 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 vn-1(xn) is a unit in k[xn] (or k[[xn]]).Proof.
If allq(n-1,i)=0, so all the k-derivations in 𝔇 stabilize the nonzero ideal 〈xn〉; in this case, R is not 𝔇-simple. Then we assume that some q(n-1,i)≠0. If vn-1(xn) is not a unit, each δi stabilizes the nontrivial ideal 〈vn-1(xn)〉. Therefore R is not 𝔇-simple.
Conversely, assume thatvn-1(xn) is a unit and notice that, in this case, there are polynomials ri(xn) such that
(5)∑i=1s‍ri(xn)q(n-1,i)(xn)=vn-1(xn),
multiplicand by the inverse of vn-1(xn), we may assume that ∑i=1sri(xn)q(n-1,i)(xn)=1. Without loss of generality, let δ1=∂x1,…,δn-1=∂xn-1; thus,
(6)∑i=ns‍ri(xn)q(n-1,i)(xn)=1.
Let I be a 𝔇-ideal. Then since I is stabilized by each δi, I is stabilized by ri(xn)δi and, then, by the k-derivation
(7)∑i=ns‍ri(xn)δi=(∑i=ns‍ri(xn)pi(xn))∂x1+⋯+(∑i=ns‍ri(xn)q(n-1,i)(xn))∂xn.
Therefore, I is stabilized by ∂x1,…,∂xn-1 and a k-derivation of the form
(8)u1(xn)∂x1+⋯+un-1(xn)∂xn-1+∂xn
for ui(xn)∈k[xn](k[[xn]]). 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 thek-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.
Let𝔇 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 elementδi of 𝔇 has the form δi=h1i(xn)∂x1+⋯+hn-1i(xn)∂xn-1, for some hji(xn)∈k[xn] (or k[[xn]]).(b)
There existsv1(xn),…,vn(xn)∈k[xn] (or k[[xn]]), such that for each δi∈𝔇 there are scalars λi,c1i,…,cn-1i∈k such that
(9)δi=∑l=1n-1‍(λivl(xn)+cli)∂xl+λivn(xn)∂xn.If, in addition,R is 𝔇-simple, then vn(xn) must be some nonzero scalar β and, also, some λj is not zero. In this case, R is also simple under the subset
(10){∂x1,…,∂xn-1,∑l=1n-1‍(λjvl(xn)+clj)∂xl+λjβ∂xn}.Proof.
If either of the conditions is met, it is clear that allk-derivations of 𝔇 will commute.
Conversely, let𝔇={δ1,…,δm} be according to the hypotheses of the theorem. By Lemma 1, each δi is of the form δi=pi(xn)∂x1+q(1,i)(xn)∂x2+⋯+q(n-1,i)(xn)∂xn.
If allq(n-1,i)(xn) are zero, then the first case holds. So, without loss of generality, we assume that q(n-1,1)(xn)≠0 and observe that
(11)δi(δ1(xn))=δi(q(n-1,1)(xn))=q(n-1,i)(xn)∂xn(q(n-1,1)(xn)),δ1(δi(xn))=δ1(q(n-1,i)(xn))=q(n-1,1)(xn)∂xn(q(n-1,i)(xn)).
Sinceδi and δ1 commute,
(12)q(n-1,i)(xn)∂xn(q(n-1,1)(xn))=q(n-1,1)(xn)∂xn(q(n-1,i)(xn)).
Then, this equation must also hold in the ring of fractions; hence we deduce that
(13)q(n-1,i)(xn)∂xn(q(n-1,1)(xn))-q(n-1,1)(xn)∂xn(q(n-1,i)(xn))(q(n-1,1)(xn))2=0.
In other words,
(14)(q(n-1,i)(xn)q(n-1,1)(xn))′=0.
Then there is someλi∈k such that q(n-1,i)(xn)=λiq(n-1,1)(xn).
Now, we observe that(15)δi(δ1(x1))=δi(p1(xn))=q(n-1,i)(xn)∂xn(p1(xn)),δ1(δi(x1))=δ1(pi(xn))=q(n-1,1)(xn)∂xn(pi(xn)).
Sinceδi and δ1 commute and both k[xn] and k[[xn]] are domains,
(16)λi∂xn(p1(xn))=∂xn(pi(xn)).
Then there is somec1i∈k such that pi(xn)=λip1(xn)+c1i. Finally, making the same argument for the other variables, we prove the desired result.
Now we suppose, in addition, thatR is 𝔇-simple. By Lemma 2, the greatest common divisor of q(n-1,1)(xn),…,q(n-1,m)(xn) must be a unit. However, we have demonstrated that all the q(n-1,i)(xn) are scalar multiples; then at least one of the q(n-1,i)(xn) must be a unit; we assume that q(n-1,j)(xn) is a unit. Since I is stabilized by
(17){(xn)∂xn∂x1,…,∂xn-1,pj(xn)∂x1+q(1,j)(xn)∂x2+⋯+q(n-1,j)(xn)∂xn},I must be trivial because, in this case, I is stabilized by ∂x1,…,∂xn-1,∂xn. Therefore, R is {∂x1,…,∂xn-1,pj(xn)∂x1+q(1,j)(xn)∂x2+⋯+q(n-1,j)(xn)∂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 Derk(k[x1,…,xn]) is a special k-derivation. This means that the divergence d* of d is 0. Moreover, it is easy to prove that the set
(18){∂x1,…,∂xn-1,∑l=1n-1‍(λjvl(xn)+clj)∂xl+λjβ∂xn}
obtained by the previous theorem is a commutative base of Derk(k[x1,…,xn]). Thus, in particular, ∑l=1n-1(λjvl(xn)+clj)∂xl+λjβ∂xn is a special derivation. However, this is easily verified since
(19)d*=∑l=1n-1‍∂xl(λjvl(xn)+clj)+∂xn(λjβ)=0.Corollary 5.
Let𝔇 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,…,fn∈R such that d(fn)=1 and d(fi)=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 isd∈𝔇 of the form
(20)d=∑l=1n-1‍(λjvl(xn)+clj)∂xl+λjβ∂xn
such that R is {∂x1,…,∂xn-1,d}-simple. Since β and λj, in the theorem, are nonzero scalars, we denote fn=(λjβ)-1xn; thus d(fn)=1.
Letfl*∈k[xn] (or k[[xn]]) such that d(fl*)=λjvl(xn)+clj. Since c(k)=0, fl* exist. Then, let fl=xl-fl*; hence d(fl)=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 ofDerk(R) However, in our case the proof is more evident (see [3, Theorem  2] (Now86)).For the remainder of this note we assume thatR is the ring k[x1,…,xn] 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,…,xn] is called locally nilpotent if for each f∈R exists a natural number n such that dn(f)=0 and we say that a basis {d1,…,dn} of Derk(k[x1,…,xn]) is locally nilpotent if every derivation di is locally nilpotent for i=1,…,n.We remember also that the Jacobian conjecture states that ifF=(F1,…,Fn) 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]).
LetR=k[x1,…,xn] be the polynomial ring in n variables over k. The following conditions are equivalent.(1)
The Jacobian conjecture is true in then-variable case.(2)
Every commutative basis of theR-module Derk(R) is locally nilpotent.(3)
Every commutative basis of theR-module Derk(R) is locally finite.Corollary 8.
Let𝔇 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 Derk(R). In particular, d is a k-derivation locally nilpotent.Proof.
The proof is immediate consequence of [3, Theorem  5].Question.
A ring is calledw-differentially simple if it is a simple relative to a family with w derivations. Recall that we are assuming R=k[x1,…,xn]; 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?



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