ISRN Algebra

Volume 2013 (2013), Article ID 560648, 4 pages

http://dx.doi.org/10.1155/2013/560648

Research Article

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

Rene Baltazar

Universidade Federal do Rio Grande do Sul (UFRGS), Campus do Vale, Código Postal 15051, 91501-970 Porto Alegre, RS, Brazil

Received 8 October 2013; Accepted 24 November 2013

Academic Editors: H. Chen and A. Zimmermann

Copyright © 2013 Rene Baltazar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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

1. Introduction

Let be a field of characteristic zero and denote either the ring of polynomials over or the ring of formal power series over .

A -derivation of is a -linear map such that for any . Denoting by the set of all -derivations of , let be a nonempty family of -derivations. An ideal of is called -stable if for all . For example, the ideals 0 and are always -stable. If has no other -stable ideal it is called -simple. When , 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 over any field of characteristic zero. More explicitly, if two derivations of are linearly independent over and commute, then they have a common Darboux polynomial or they are Jacobian derivations; in [1], the authors proved the same result for and . 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 is equivalent to the assertion that every commutative basis of is locally nilpotent.

Let be a set of commute -derivations of ; then is -simple if and only if it is -simple for some -derivation (see [5, Corollary  2.10]). In , as pointed out in [5], up to scalar multiples, these are only sets of two commuting, nonsimple -derivations such that both and is -simple. Motivated by this, we analyze this result in [5] for and then we propose some connections with known results on commutative bases of derivations in . More precisely, the derivations are not simple -derivations of ; however, as will be shown, they can be part of a set of commuting, nonsimple -derivations such that is -simple. A trivial example is . Using the notations in [3], we give a nontrivial commutative base containing only nonsimple -derivations of the free -module such that is -simple and if the Jacobian conjecture is true in , 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 -derivations of that commute with is

such that ,   (or if ).

Proof. It is clear that all derivations of this form commute with the derivations . For the converse, let

be a -derivation of that commutes with . Then

for all . Thus, . Similarly, we can prove that .

Let be any finite set of -derivation of that commutes with but not necessarily with each other. By Lemma 1, each is of the form

We denote by the greatest common divisor of . We have the following characterization for the simplicity of .

Lemma 2. Using the notations above, is -simple if and only if is a unit in (or ).

Proof. If all , so all the -derivations in stabilize the nonzero ideal ; in this case, is not -simple. Then we assume that some . If is not a unit, each stabilizes the nontrivial ideal . Therefore is not -simple.

Conversely, assume that is a unit and notice that, in this case, there are polynomials such that

multiplicand by the inverse of , we may assume that . Without loss of generality, let ; thus,

Let be a -ideal. Then since is stabilized by each , is stabilized by and, then, by the -derivation

Therefore, is stabilized by and a -derivation of the form

for . Thus is stabilized by and, then, we deduce that must be a trivial ideal.

Note that until now we only assume that all the -derivations commute with not that all elements commute with each other. Using the previous lemmas, the following theorem will show that if is -simple under a set of commuting -derivations that contains , then is simple under a subset of commuting nonsimple -derivations.

Theorem 3. Let be a set of -derivations of such that . Then the derivations of commute with each other if and only if one of the following two cases holds.

(a)Each element of has the form , for some (or ).(b)There exists (or ), such that for each there are scalars such that

If, in addition, is -simple, then must be some nonzero scalar and, also, some is not zero. In this case, is also simple under the subset

Proof. If either of the conditions is met, it is clear that all -derivations of will commute.

Conversely, let be according to the hypotheses of the theorem. By Lemma 1, each is of the form .

If all are zero, then the first case holds. So, without loss of generality, we assume that and observe that

Since and commute,

Then, this equation must also hold in the ring of fractions; hence we deduce that

In other words,

Then there is some such that .

Now, we observe that

Since and commute and both and are domains,

Then there is some such that . Finally, making the same argument for the other variables, we prove the desired result.

Now we suppose, in addition, that is -simple. By Lemma 2, the greatest common divisor of must be a unit. However, we have demonstrated that all the are scalar multiples; then at least one of the must be a unit; we assume that is a unit. Since is stabilized by

must be trivial because, in this case, is stabilized by . Therefore, is -simple which completes the proof.

Remark 4. Nowicki in [7, Theorem  2.5.5] proved that every -derivation of a commutative base of is a special -derivation. This means that the divergence of is . Moreover, it is easy to prove that the set

obtained by the previous theorem is a commutative base of . Thus, in particular, is a special derivation. However, this is easily verified since

Corollary 5. Let be a set of commute -derivations of such that is -simple and . Then, there exists and there exist elements such that and , for any , so that is -simple.

Proof. By the previous theorem, we know that there is of the form

such that is -simple. Since and , in the theorem, are nonzero scalars, we denote ; thus .

Let (or ) such that . Since , exist. Then, let ; hence , for any . This completes the proof.

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

For the remainder of this note we assume that is the ring of polynomials over and is as in the previous theorem and also we recall the following definitions.

We recall from [7] that a -derivation of is called locally nilpotent if for each exists a natural number such that and we say that a basis of is locally nilpotent if every derivation is locally nilpotent for .

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

Theorem 7 (see [3, Theorem  5]). Let be the polynomial ring in variables over . The following conditions are equivalent.

(1)The Jacobian conjecture is true in the -variable case.(2)Every commutative basis of the -module is locally nilpotent.(3)Every commutative basis of the -module is locally finite.

Corollary 8. Let be a set of commute -derivations of such that is -simple, , and the Jacobian conjecture is true in . Then, there exists such that is a locally nilpotent commutative base of the -module . In particular, is a -derivation locally nilpotent.

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

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

Acknowledgment

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

References

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