The Degrees of Nilpotency of Nilpotent Derivations on the Ring of Matrices

Transcription

1 Internatonal Mathematcal Forum, Vol. 6, 2011, no. 15, The Degrees of Nlpotency of Nlpotent Dervatons on the Rng of Matrces Homera Pajoohesh Department of of Mathematcs Medgar Evers College of CUNY Brooklyn, NY , USA Melssa Vanze-Butler Department of of Mathematcs Medgar Evers College of CUNY Brooklyn, NY , USA Abstract We consder M n (R, the rng of n by n matrces on the real numbers where n 2. We fnd all of the natural numbers that are the degree of nlpotency of some nlpotent dervatons on M n (R. Mathematcs Subject Classfcaton: 16S50, 13N15, 16N40 Keywords: dervatons Rng of matrces, dervatons, nner dervatons, nlpotent 1 Introducton Dervatons on rngs help us to understand rngs better and also dervatons on rngs can tell us about the structure of the rngs. For nstance a rng s commutatve f and only f the only nner dervaton on the rng s zero. Also dervatons can be helpful for relatng a rng wth the set of matrces wth entres n the rng (see, [19]. Dervatons play a sgnfcant role n determnng whether a rng s commutatve, see [11], [3] and [1]. Dervatons can also be useful n other felds. For example, dervatons play a role n the calculaton of the egenvalues of matrces (see, [2] whch s mportant n mathematcs and other scences, busness and engneerng. Dervatons also are used n quantum

2 714 H. Pajoohesh and M. Vanze-Butler physcs(see, [18]. Dervatons can be added and subtracted and we stll get a dervaton, but when we compose a dervaton wth tself we do not necessarly get a dervaton. For nstance n Example 2.12, ( d 2 A s not a dervaton because d 2 A (XY d2 A (XY +Xd2 A (Y for X Y. In 1957 n [20] Posner 1 0 proved that f d 1 and d 2 are two non-zero dervatons on a prme rng whose characterstc s not 2, then d 1 d 2 s not a dervaton. Thus n a prme rng R whose characterstc s not 2 f d 2 s a dervaton then d must be zero. In partcular when d 2 s the zero dervaton then d 0. Ths means that the only nlpotent dervaton wth degree of nlpotency 2 on a prme rng whose characterstc s not 2 s the zero dervaton. In other words there s no non-zero nlpotent dervaton wth degree of nlpotency 2 on a prme rng whose characterstc s not 2. An example of such a rng s M n (R. But, sometmes when we compose a dervaton a few tmes (more than two tmes wth tself on these knd of rngs we get the zero dervaton, for nstance n Example 2.12, d A s a non-zero dervaton whch s nlpotent wth the degree of nlpotency 3. But, sometmes condtons on the rng and dervaton forces the nlpotent dervaton to be zero, see [7]. In ths project we work on nlpotent dervatons on M n (R. Frst, we consder some nlpotent nner dervatons on M n (R and we fnd ther degrees of nlpotences. Then we use several results to fnd all of the possble degrees of nlpotences of nlpotent dervaton on M n (R n general. To learn more about nlpotent dervatons and dervatons on matrces see [5], [16] and [19]. 2 Prelmnary Notes The defntons n ths secton can be found n many rng theory books for example [15] and [13]. Defnton 2.1 An element x n a rng R s called nlpotent f there s a natural number n such that x n 0and the smallest natural number wth ths property s called the degree of nlpotency of x. ( 0 1 Example 2.2 The matrx A M n (R s nlpotent wth the degree of nlpotency 2 because A 2 0and A 1 A 0. Defnton 2.3 A rng R s called prme provded that arb 0mples a 0or b 0for every, a, b R. Defnton 2.4 A rng R s called semprme provded that ara 0mples a 0for every, a R.

3 The degrees of nlpotency of nlpotent dervatons 715 Defnton 2.5 A rng R s called smple f the only two sded deals of R are tself and the zero deal. By an deal of R we mean a nonempty subset S of R such that for every x, y S and r R we have x y, rx S. Example 2.6 The rng of real numbers s a prme, semprme and smple. It s known that f a rng S s prme then M n (S, the set of n by n matrces wth entres n S, s a prme rng, see [13] page 449. Thus M n (R s a prme rng and therefore t s semprme. Also, t has been shown n [9] that M n (R s smple. Defnton 2.7 If n s a natural number we say that the characterstc of a rng S s n f for every x S we have nx 0and f such n does not exst we say the characterstc of S s zero. It s clear that the characterstc of M n (R s zero. The usual dervatve operator s addtve and (fg f g + fg. Ths defnton has been generalzed for every rng as follows. Defnton 2.8 For a rng R, an addtve map d : R R s called a dervaton on R f t satsfes the product rule d(ab d(ab + ad(b. Example 2.9 Consder the rng R R[x] of all real polynomals. Then the usual dervatve of real polynomals s a dervaton on R. Defnton 2.10 For a n a rng R, d a : R R s defned by d a (x [a, x] where, [a, x] ax xa. It s easly verfed that d a s a dervaton on R, and such dervatons are called nner dervatons. It s clear that a rng R s commutatve f and only f every nner dervaton on t s zero. Defnton 2.11 Let d be a dervaton on the rng R. For n>0, the n th dervaton of d s denoted by d n and s obtaned when d s composed wth tself n tmes, and by d 0 (x we mean the dentty functon defned by d 0 (x x. A dervaton d on R s nlpotent f, for some natural number n, d n (x 0for every x R and the smallest natural number wth ths property s called the degree of nlpotency of d. Example 2.12 Consder d A on M 2 (R, where A ( x y X, we have z t d A (X AX XA ( 0 1. Now for ( ( ( ( 0 1 x y x y 0 1 z t z t ( ( ( z t 0 x z t x 0 z 0 z

4 716 H. Pajoohesh and M. Vanze-Butler Thus, d 2 A (X d A(d A (X Ad A (X d A (XA ( ( ( 0 1 z t x z t x 0 z 0 z ( ( ( 0 z 0 z 0 2z Thus ( 0 1 d 3 A(X Ad 2 A(XA d 2 ( A(XA ( ( z 0 2z ( ( ( ( 0 1 ( Therefore d 3 A 0and also d2 A 0because for example d A( 1 0 Ths proves that the degree of nlpotency of d A on M 2 (R s three. 0. Remark 2.13 For any rng wth dervaton d and for any non-negatve nteger n, d n (xy n ( n 0 d (xd n (y s the expanson of the n th dervaton of a product. 3 Some odd degrees of nlpotences In ths secton we dscuss some of the degrees of nlpotences on M n (R, where n 2. We denote by δ,j the matrx n M n (R whose entry n the th row and j th column s 1 and all other entres are 0. It follows that δ,j δ p,q δ,q f j p otherwse t s zero. In the next theorem we try to generalze the Example 2.12 n general for M n (R. Theorem 3.1 There s a nlpotent nner dervaton wth degree of nlpotency 3 for M n (R, where n 2. Proof. Consder d : M n (R M n (R such that d(x δ 1,n X Xδ 1,n. We need to prove that d s nlpotent wth degree 3. If we compose d wth tself, we get d 2 (X d(d(x δ 1,n d(x d(xδ 1,n δ 1,n (δ 1,n X Xδ 1,n (δ 1,n X Xδ 1,n δ 1,n δ 1,n δ 1,n X δ 1,n Xδ 1,n δ 1,n Xδ 1,n +Xδ 1,n δ 1,n (0X 2δ 1,n Xδ 1,n +

5 The degrees of nlpotency of nlpotent dervatons 717 X(0 2δ 1,n Xδ 1,n. Thus d 2 (x 2δ 1,n Xδ 1,n. If we compose d wth d 2, we get d 3 (X d(d 2 (X δ 1,n d 2 (X d 2 (Xδ 1,n δ 1,n ( 2δ 1,n Xδ 1,n ( 2δ 1,n Xδ 1,n δ 1,n 2δ 1,n δ 1,n Xδ 1,n +2δ 1,n Xδ 1,n δ 1n 2(0Xδ 1,n +2δ 1,n X(0 0Thusd 3 0 but d 2 0 because for example d 2 (δ n,1 0. Ths proves that, d s nlpotent wth degree of nlpotency 3. In the next theorem we show that we can get more odd degrees of nlpotences. Theorem 3.2 For every odd number k between 3 and 2n 1, nclusve, there s a nlpotent dervaton wth degree of nlpotency k on M n (R, where n 2. Proof. Let k 2j 1. Defne d : M n (R M n (R such that d(x AX XA, where A j 1 1 δ,+1. We wll use nducton to prove that d t (X t ( t 0 ( 1 t A t XA. When t 0,d 0 (X X and 0 ( 0 0 ( 1 0 A 0 XA X. We assume ths s true for k 1 and we wll show that t s true for k. Notce that d k (X Ad k 1 (X d k 1 (XA ( k 1 ( k 1 A( ( 1 A k 1 XA ( ( 1 A k 1 XA A 0 0 ( k 1 ( k 1 ( 1 A k XA ( 1 A k 1 XA +1 (1 0 0 ( k 1 k ( k 1 ( 1 A k XA ( 1 1 A k XA ( k 1 k ( k 1 ( 1 A k XA + ( 1( 1 1 A k XA. ( In (1 we change the ndex from to 1. In order to make the summatons n (2 the same, we need to wrte out the frst term when 0 for the frst sum and the last term when k for the second sum. Thus, the equaton

6 718 H. Pajoohesh and M. Vanze-Butler becomes: ( k 1 ( k 1 d k (X ( 1 0 A k 0 XA 0 + ( 1 A k XA 0 1 k ( k 1 + ( 1 A k XA +( 1 k XA k 1 1 ( ( k 1 k 1 A k X + ( 1 ( + A k XA +( 1 k XA k 1 1 ( k A k X + ( 1 A k XA +( 1 k XA k 0 1 k ( k ( 1 A k XA. Thus, d t (X t 0 ( t ( 1 A t XA. (3 Now, we show that f A j 1 1 δ,+1 then A s nlpotent and ts degree of nlpotency s j. Note that A 2 j 1 1 δ j 1,+1 1 δ,+1 j 2 1 δ,+2. Smlarly A 3 j 3 1 δ,+3 and n general A k j k 1 δ,+k. So that A j δ,+(j 1 δ 1,j and A j 0. Lookng back at the equaton d t (X t ( t 0 ( 1 t A t XA. When t 2j 1, then t j or j. Thus A t 0orA 0. Therefore d 2j 1 (x 0. Also, note that snce A k 0 for k j, d 2j 2 (X 2j 2 ( 2j 2 0 ( 1 A t XA ( 2j 2 j 1 ( 1 j 1 δ 1,j Xδ 1,j. Thus, d 2j 2 (δ j,1 0. Ths proves that the degree of nlpotency of d s 2j 1. 4 All degrees of nlpotences of nlpotent dervatons In the prevous secton for every n>1 we found nlpotent nner dervatons on M n (R wth odd degree of nlpotency from 3 to 2n 1. For example there are nlpotent nner dervatons wth degrees of nlpotency 3, 5, 7, 9, 11, 13 on M 7 (R. In ths secton we characterze, for each n>1, the natural numbers that are a degree of nlpotency of a nlpotent dervaton of M n (R. We recall that for a natural number k, a rng R s called k-torson free f kx 0 for x R mples x 0 and a rng s torson free f t s k-torson free

7 The degrees of nlpotency of nlpotent dervatons 719 for every natural number k. One can easly see that M n (R s torson free and therefore 2-torson free. Chung and Luh n [6] proved that the nlpotency of a dervaton on a 2-torson free semprme rng s always an odd number. Thus, we can apply the result of Chung and Luh and we conclude that: Theorem 4.1 If d s a nlpotent dervaton on M n (R then ts degree of nlpotency must be an odd number. It has been shown by Kharchenko n [14] that when d s a nlpotent dervaton on a prme rng S wth degree of nlpotency n, then d s nner by an element n ts Martndale quotent rng Q, when the characterstc of R s zero or exceeds n. Ths means that there s an element t Q such that for every x S, d(x tx xt. It wll be seen that there s no need n ths partcular paper to know detals about the Martndale quotent rng. But nformaton and detals about the Martndale quotent rng can be found n [8] and [17]. Kharchenko s result was mproved n [4]. There Chuang and Lee proved that f d s a nlpotent dervaton on the rng R, where R s prme and ts characterstc s zero then d s nner by a nlpotent element n the Martndale quotent rng. On the other hand, n 1988 n [10] Hersten proved that f S s a smple rng wth dentty then Q, ts Martndale rng of quotents, s S tself. Thus, the Martndale quotent rng of M n (R sm n (R. The rng M n (R s a prme and ts characterstc s zero. Thus, f we apply Chuang and Lee s theorem and Hersten s theorem for M n (R, we conclude that: Theorem 4.2 If d s a nlpotent dervaton on M n (R then d s nner by a nlpotent element n M n (R. Thus, f d s a nlpotent dervaton on M n (R then there s a nlpotent element t n M n (R such that for every x M n (R we have, d(x tx xt. In [12] n Problem 3, page 139 they gve a hnt for showng that the degree of nlpotency of an n by n nlpotent matrx s always less than or equal to n. In Theorem 3.2, we showed that f A s a nlpotent matrx n M n (R wth degree of nlpotency l, then the nner dervaton correspondng to A s nlpotent wth the degree of nlpotency 2l 1. Thus we can say: Man result: If d s a nlpotent dervaton on M n (R whose the degree of nlpotency s t, then t s an odd number such that 3 t 2n 1. Moreover, for every odd number t wth 3 t 2n 1, there s nlpotent dervaton on M n (R whose degree of nlpotency s t.

8 720 H. Pajoohesh and M. Vanze-Butler Remark 4.3 One can use the references and verfy that our man result stll s vald f we replace R wth any torson free feld. ACKNOWLEDGEMENTS. Ths research was supported by LSAMP, The New York Cty Lous Stokes Allance for Mnorty Partcpaton. The authors are thankful to Professor Charles Lansk for hs valuable nformaton and to Dr. Bart Van Sterteghem for useful dscussons References [1] S. Andma and H. Pajoohesh, Commutatvty of prme rngs wth dervatons, Acta Math. Hungarca, Vol. 128 (1-2 (2010, [2] G.A. Baker, A new dervaton of Newton s denttes and ther applcaton to the calculaton of the egenvalues of a matrx. J. Soc. Indust. Appl. Math. 7 ( [3] H. E. Bell and M. N. Daf, On dervatons and commutatvty n prme rngs. Acta Math. Hungar. Vol. 66 (1995, [4] C. L. Chuang and T.K. Lee, Nlpotent dervatons, Journal of Algebra, 287 (2005, [5] L. O. Chung and J. Luh, Nlpotency of dervatons on an deal, Proc. Amer. Math. Soc. 90 (1984, [6] L. O. Chung and J. Luh, Nlpotency of dervatons, Canad. Math. Bull, 26(3, (1983, [7] M. Ebrahm and H. Pajoohesh, Composton of dervatons on (semprme l-rngs H. Kyungpook Math Journal. 44 (2004, no. 2, [8] V.K. Harcenko, The Galos theory of semprme rngs., Algebra Logka, 16 (1977, no. 3, [9] D.W. Henderson, A short proof of Wedderburn s theorem. Amer. Math. Monthly, 72 (1965, [10] I.N. Hersten, A condton that a dervaton be nner. Rend. Crc. Mat. Palermo (2 37 (1988, no. 1, 5-7. [11] I. N. Hersten, A note on dervatons. Canad. Math. Bull, 21 (1978,

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