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

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1 ISRN Agebra Voume 2013 Artice ID page Reearch Artice Simpicity and Commutative Bae of Derivation in Poynomia and Power Serie Ring Rene Batazar Univeridade Federa do Rio Grande do Su (UFRGS) Campu do Vae Código Pota Porto Aegre RS Brazi Correpondence houd be addreed to Rene Batazar; rene.batazar@ufrg.br Received 8 October 2013; Accepted 24 November 2013 Academic Editor: H. Chen and A. Zimmermann Copyright 2013 Rene Batazar. Thi i an open acce artice ditributed under the Creative Common Attribution Licene which permit unretricted ue ditribution and reproduction in any medium provided the origina work i propery cited. The firt part of the paper wi decribe a recent reut of Retert in (2006) for k[x 1...x n ] and k[[x 1...x n ]]. Thi reut tate that if D i a et of commute k-derivation of k[x y] uch that both x D and the ring i D-impe then there i d D uch that k[x y] i { x d}-impe. A for appication we obtain reationhip with known reut of A. Nowicki on commutative bae of derivation. 1. Introduction Let k be a fied of characteritic zero and R denote either the ring k[x 1...x n ] of poynomia over k or the ring k[[x 1...x n ]] of forma power erie over k. A k-derivation d:r Rof R i a k-inear map uch that d(ab) = d(a)b + ad(b) for any a b R. Denotingby Der k (R) the et of a k-derivation of RetD Der k (R) be anonemptyfamiyofk-derivation. An idea I of R i caed D-tabe if d(i) I for a d D. For exampe the idea 0 and R are away D-tabe. If R ha no other D-tabe idea it i caed D-impe.WhenD ={d} d i often caed a impe derivation. The commuting derivation have been tudied by evera author: Li and Du [1] Maubach [2] Nowicki [3] Petravchuk [4] Retert [5] Van den Een [6]. For exampeitiweknownthateachpairofcommutinginear operator on a finite dimeniona vector pace over an agebraicay coed fied ha a common eigenvector; in [4] Petravchuk proved an anaogou tatement for derivation of k[x y] over any fied k of characteritic zero. More expicity if two derivation of k[x y] are ineary independent over k and commute then they have a common Darboux poynomia or they are Jacobian derivation; in [1] the author proved the ame reut for k[x 1...x n ] and k[[x 1...x n ]]. However we oberve that thi reut ha aready been proved by Nowicki in (Now86) for both ring. Another intereting reut wa proved by Nowicki in [3 Theorem 5] which ay that the famou Jacobian conjecture in k[x 1...x n ] i equivaent to the aertion that every commutative bai of Der k (R) i ocay nipotent. Let D be a et of commute k-derivation of k[x]; then k[x] i D-impe if and ony if it i d-impe for ome k-derivation d D (ee [5 Coroary2.10]).Ink[x y] a pointed out in [5] up to caar mutipe thee are ony et D of two commuting nonimpe k-derivation uch that both x D and R i D-impe. Motivated by thi we anayze thi reut in [5] forr andthenwe propoe ome connection with known reut on commutative bae of derivation in R. More preciey the derivation are not impe k-derivation of R; however a wi be hown they can be part of a et D of n commuting nonimpe k-derivation uch that R i D-impe. A trivia exampe i D =... xn }.Uing the notation in [3] we give a nontrivia commutative bae containing ony nonimpe k-derivation of the free R-modue Der k (R) uch that R i D-impe and if the Jacobian conjecture i true in k[x 1...x n ]aaconequence of [3 Theorem5]weobtainafamiyofocaynipotent derivation.

2 2 ISRN Agebra 2. Commuting Derivation and Simpicity Lemma 1. The et of a k-derivation of R that commute with i {p (x n ) +q 1 (x n ) x2 + +q (x n ) xn } (1) uch that p q 1...q k[x n ] (or k[[x n ]] if R = k[[x 1... x n ]]). Proof. It i cear that a derivation of thi form commute with the derivation. For the convere et d =p (x 1...x n ) +q 1 (x 1...x n ) x2 + +q (x 1...x n ) xn (2) be a k-derivation of R that commute with.then 0=d ( xi (x 1 )) = xi (d (x 1 )) = xi (p (x 1...x n )) (3) for a i = 1...Thup (x 1...x n ) k[x n ](k[[x n ]]). Simiary we can prove that q i (x 1...x n ) k[x n ](k[[x n ]]). Let D ={δ 1...δ } be any finite et of k-derivation of R that commute with but not neceariy with each other. By Lemma 1 each δ i i of the form δ i =p i (x n ) +q (1i) (x n ) x2 + +q (i) (x n ) xn. (4) We denote by V (x n ) the greatet common divior of q (1)...q (). We have the foowing characterization for the impicity of R. Lemma 2. Uing the notation above R i D-impe if and ony if V (x n ) i a unit in k[x n ] (or k[[x n ]]). Proof. If a q (i) =0oathek-derivation in D tabiize the nonzero idea x n ;inthicaer i not D-impe. Then we aume that ome q (i) =0.IfV (x n ) i not a unit each δ i tabiize the nontrivia idea V (x n ). Therefore R i not D-impe. Converey aume that V (x n ) i a unit and notice that in thi cae there are poynomia r i (x n ) uch that i=1 r i (x n )q (i) (x n )=V (x n ) (5) mutipicand by the invere of V (x n ) we may aume that i=1 r i(x n )q (i) (x n )=1. Without o of generaity et δ 1 =...δ = x ;thu r i (x n )q (i) (x n )=1. (6) Let I be a D-idea. Then ince I i tabiized by each δ i I i tabiized by r i (x n )δ i and then by the k-derivation r i (x n )δ i =( + +( r i (x n )p i (x n )) r i (x n )q (i) (x n )) xn. Therefore I i tabiized by and a k-derivation of the form u 1 (x n ) + +u (x n ) x + xn (8) for u i (x n ) k[x n ](k[[x n ]]). Thu I i tabiized by xn and then we deduce that I mut be a trivia idea. Note that unti now we ony aume that a the kderivation commute with not that a eement commute with each other. Uing the previou emma the foowing theorem wi how that if R i D-impe under a et D of commuting k-derivation that contain then R i impe under a ubet of n commuting nonimpe kderivation. Theorem 3. Let D be a et of k-derivation of R uch that D. Then the derivation of D commute with each other if and ony if one of the foowing two cae hod. (a) Each eement δ i of D ha the form δ i =h i 1 (x n) + + h i (x n) x foromeh i j (x n) k[x n ] (or k[[x n ]]). (b) There exit V 1 (x n )...V n (x n ) k[x n ] (or k[[x n ]]) uch that for each δ i D there are caar λ i c i 1...ci kuch that δ i = (7) (λ i V (x n )+c i ) x +λ i V n (x n ) xn. (9) If in addition R i D-impe then V n (x n ) mut be ome nonzero caar β and ao ome λ j i not zero. In thi cae R i ao impe under the ubet (λ j V (x n )+c j ) x }. (10) Proof. If either of the condition i met it i cear that a kderivation of D wi commute. Converey et D = {δ 1...δ m } be according to the hypothee of the theorem. By Lemma 1 each δ i i of the form δ i =p i (x n ) +q (1i) (x n ) x2 + +q (i) (x n ) xn. If a q (i) (x n ) are zero then the firt cae hod. So without o of generaity we aume that q (1) (x n ) =0and oberve that δ i (δ 1 (x n )) = δ i (q (1) (x n )) =q (i) (x n ) xn (q (1) (x n )) δ 1 (δ i (x n )) = δ 1 (q (i) (x n )) =q (1) (x n ) xn (q (i) (x n )). (11)

3 ISRN Agebra 3 Since δ i and δ 1 commute q (i) (x n ) xn (q (1) (x n )) (12) =q (1) (x n ) xn (q (i) (x n )). Then thi equation mut ao hod in the ring of fraction; hence we deduce that q (i) (x n ) xn (q (1) (x n )) q (1) (x n ) xn (q (i) (x n )) (q (1) (x n )) 2 =0. (13) In other word ( q (i) (x n ) q (1) (x n ) ) =0. (14) Then there i ome λ i k uch that q (i) (x n ) = λ i q (1) (x n ). Now we oberve that δ i (δ 1 (x 1 )) = δ i (p 1 (x n )) =q (i) (x n ) xn (p 1 (x n )) δ 1 (δ i (x 1 )) = δ 1 (p i (x n )) =q (1) (x n ) xn (p i (x n )). (15) Since δ i and δ 1 commute and both k[x n ] and k[[x n ]] are domain λ i xn (p 1 (x n )) = xn (p i (x n )). (16) Then there i ome c i 1 kuch that p i(x n )=λ i p 1 (x n )+c i 1. Finaymakingtheameargumentfortheothervariabewe prove the deired reut. Now we uppoe in addition that R i D-impe. By Lemma 2 the greatet common divior of q (1) (x n )...q (m) (x n ) mut be a unit. However we have demontrated that a the q (i) (x n ) are caar mutipe; then at eat one of the q (i) (x n ) mut be a unit; we aume that q (j) (x n ) i a unit. Since I i tabiized by p j (x n ) +q (1j) (x n ) x2 + +q (j) (x n ) xn } (17) I mut be trivia becaue in thi cae I i tabiized by xn. Therefore R i p j (x n ) + q (1j) (x n ) x2 + + q (j) (x n ) xn }-impe which compete the proof. Remark 4. Nowicki in [7 Theorem 2.5.5] proved that every k-derivation of a commutative bae of Der k (k[x 1...x n ]) i apeciak-derivation. Thi mean that the divergence d of d i 0.Moreoveritieaytoprovethattheet (λ j V (x n )+c j ) x } (18) obtained by the previou theorem i a commutative bae of Der k (k[x 1...x n ]). Thuinparticuar (λ jv (x n )+ c j ) x i a pecia derivation. However thi i eaiy verified ince d = x (λ j V (x n ) +c j ) + x n (λ j β) =0. (19) Coroary 5. Let D be a et of commute k-derivation of R uch that R i D-impe and D. Thenthere exit d Dand there exit eement f 1...f n Ruch that d(f n )=1and d(f i )=0foranyi = 1...othatR i d}-impe. Proof. By the previou theorem we know that there i d D of the form d= (λ j V (x n )+c j ) x (20) uch that R i d}-impe. Since β and λ j inthe theorem are nonzero caar we denote f n =(λ j β) 1 x n ;thu d(f n )=1. Let f k[x n ] (or k[[x n ]])uchthatd(f )=λ j V (x n )+ c j.sincec(k) = 0 f exit. Then et f = x f ;hence d(f )=0forany = 1...Thicompetetheproof. Remark 6. The previou coroary i a particuar cae of an important theorem about the characterization of commutative bai of Der k (R) However in our cae the proof i more evident (ee [3 Theorem 2] (Now86)). For the remainder of thi note we aume that R i the ring k[x 1...x n ] of poynomia over k and d} i a in the previou theorem and ao we reca the foowing definition. We reca from [7] thatak-derivation d of k[x 1...x n ] i caed ocay nipotent if for each f Rexit a natura number n uch that d n (f) = 0 and we ay that a bai {d 1...d n } of Der k (k[x 1...x n ]) i ocay nipotent if every derivation d i i ocay nipotent for i=1...n. We remember ao that the Jacobian conjecture tate that if F=(F 1...F n ) i a poynomia map uch that the Jacobian matrix i invertibe then F haapoynomiainvere(ee[7]). Theorem 7 (ee [3Theorem 5]). Let R=k[x 1...x n ] be the poynomia ring in n variabe over k. The foowing condition are equivaent. (1) The Jacobian conjecture i true in the n-variabe cae. (2) Every commutative bai of the R-modue Der k (R) i ocay nipotent. (3) Every commutative bai of the R-modue Der k (R) i ocay finite. Coroary 8. Let D be a et of commute k-derivation of R uch that R i D-impe D andthejacobian

4 4 ISRN Agebra conjecture i true in R. Then there exit d D uch that d}i a ocay nipotent commutative bae of the R-modue Der k (R). Inparticuard i a k-derivation ocay nipotent. Proof. The proof i immediate conequence of [3 Theorem 5]. Quetion. Aringicaedw-differentiay impe if it i a impe reative to a famiy with w derivation. Reca that we are auming R = k[x 1...x n ];thenweknowthatr i 1- differentiay impe and dim(r)-differentiay impe a we. However n = dim(r) i not neceariy the maet w for which uch a ring can be w-differentiay impe (ee [8]). Thu one may ak the foowing: what i the maet poitive integer w =1uch that R i w-differentiay impe and a w derivation are nonimpe and commute? Acknowedgment The reearch of Rene Batazar wa partiay upported by CAPES of Brazi. Reference [1] J. Li and X. Du Pairwie commuting derivation of poynomia ring Linear Agebra and It Appication vo.436no.7pp [2] S. Maubach The commuting derivation conjecture Journa of Pure and Appied Agebravo.179no.1-2pp [3] A. Nowicki Commutative bae of derivation in poynomia and power erie ring Pure and Appied Agebravo. 40 no. 3 pp [4] A. P. Petravchuk On pair of commuting derivation of the poynomia ring in one or two variabe Linear Agebra and It Appicationvo.433no.3pp [5] K. Retert Set of commuting derivation and impicity Communication in Agebravo.34no.8pp [6] A. van den Een Poynomia Automorphim and the Jacobian Conjecture vo.190ofprogre in Mathematic Birkhäuer Bae Switzerand [7] A. Nowicki Poynomia Derivation and Their Ring of Contant N. Copernicu Univerity Pre Torun Poand [8] D. Levcovitz and S. C. Coutinho On the differentia impicity of affine ring Proceeding of the American Mathematica Society.Inpre.

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