On locally nilpotent derivations of Boolean semirings

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1 Daowsud et al., Cogent Mathematcs 2017, 4: htts://do.org/ / PURE MATHEMATICS RESEARCH ARTICLE On locally nlotent dervatons of Boolean semrngs Katthaleeya Daowsud 1, Monrudee Srvoravt 1 and Utsanee Leerawat 1 * Receved: 08 March 2017 Acceted: 13 June 2017 Frst Publshed: 06 July 2017 *Corresondng author: Utsanee Leerawat, Faculty of Scence, Deartment of Mathematcs, Kasetsart Unversty, Bangkok 10900, Thaland E-mal: fscutl@ku.ac.th Revewng edtor: Ngel Byott, Unversty of Exeter, UK Addtonal nformaton s avalable at the end of the artcle Abstract: In ths aer, we consder the comoston of dervatons n Boolean semrngs and nvestgate the condtons that the comoston of two dervatons s a dervaton. We also show that the nth dervaton of a dervaton d, denoted by d n, on a Boolean semrng satsfes Lebnz rule. Fnally, we show that any locally notent dervatons on a zero-symmetrc Boolean semrng must be zero. Subjects: Scence; Mathematcs & Statstcs; Advanced Mathematcs; Algebra Keywords: semrng; Boolean semrng; dervaton; locally notent AMS subject classfcatons: 13N15; 16W25 1. Introducton The noton of the rng wth dervaton lays a sgnfcant role n the ntegraton of analyss, algebrac geometry and algebra. By a dervaton of a rng R, we mean any functon d:r R whch satsfes the followng condtons: da + b =da+db and dab =dab + adb, for all a, b R. The study of dervatons of rme rng was ntated by Posner Posner consdered the comoston of dervatons and showed that the comoston of two nonzero dervatons of a rme rng R cannot be a dervaton rovded that characterstc of R s dfferent from 2. Bresar 1990 and Ashraf and Nadeem 2001 roved commutatvty of rme and semrme rngs wth dervatons satsfyng certan olynomal constrants. Chebotar 1995 and Bell and Argac 2001 obtaned the necessary condton when the comoston of dervatons could be a dervaton. Furthermore, Bell and Argac 2001 ABOUT THE AUTHORS Katthaleeya Daowsud s a lecturer n the Deartment of Mathematcs at Kasetsart Unversty. She receved a PhD Mathematcs n 2013 from Oregon State Unversty, USA. Her research nterests are Algebra, Number Theory, and Algebrac geometry. Monrudee Srvoravt s a lecturer n the Deartment of Mathematcs at Kasetsart Unversty. She obtaned her MS degree n Mathematcs from Chulalongkorn Unversty, Thaland n Her research nterests are Algebra and Number Theory. Utsanee Leerawat s an assocate rofessor n the Deartment of Mathematcs at Kasetsart Unversty. She receved her PhD degree n Mathematcs n 1994 at Chulalongkorn Unversty, Thaland. Her research nterests are Rng Theory dervatons, commutatvty condtons n rngs, Unversal algebra and Combnatorcs. PUBLIC INTEREST STATEMENT A dervaton s a functon on an algebra htts:// en.wkeda.org/wk/algebra_over_a_feld whch generalzes certan features of the dervatve htts://en.wkeda.org/wk/dervatve oerator. Dervatons lay a sgnfcant role n the ntegraton of analyss, algebrac geometry and algebra. The study of dervatons n rngs though ntated long back, but got nterested only after Posner who establshed two very strkng results on dervatons n rme rngs n The noton of dervaton has also been generalzed n varous drectons, such as Jordan dervaton, generalzed dervaton, generalzed Jordan dervaton etc. The objectve of ths aer s to study the comoston of dervatons on Boolean semrng. Also, we nvestgate the condtons that the comoston of two dervatons s a dervaton. Moreover, we nvestgate the nth dervaton satsfyng Lebnz rule. Fnally, we show that a locally nlotent dervatons on a Boolean semrng ossesng some condton must be zero The Authors. Ths oen access artcle s dstrbuted under a Creatve Commons Attrbuton CC-BY 4.0 lcense. Page 1 of 7

2 Daowsud et al., Cogent Mathematcs 2017, 4: htts://do.org/ / and Wang 1994 have nvestgated the nvarance of certan deals under dervatons. Lee and Lee 1986, roved that f d s a dervaton on a rme rng R wth center Z such that d n x Z for all x, where n s a fxed nteger, then ether d n x Z for all x n R or R s a commutatve ntegral doman. Boolean semrng has been used n the mathematcal lterature wth at least two dfferent meanngs. The frst one was gven by Subrahmanyam The second one was ntroduced by Galbat and Veones 1980, whch has also been nvestgated by Guzman In ths aer we use the defnton of Boolen semrng n the sense of Subrahmanyan. In ths aer our objectve s to study the comoston of dervatons on Boolean semrng Subrahmanyam, Also we nvestgate the condtons that the comoston of two dervatons s a dervaton. Moreover, we nvestgate the nth dervaton satsfyng Lebnz rule. A dervaton d of a rng R s sad to be locally nlotent f for any x R, there exsts a ostve nteger n such that d n x =0. Locally nlotent dervatons lay an mortant role n commutatve algebra and algebrac geometry, and several roblems may be formulated usng locally nlotent dervatons see Essen, 1995; Ferrero, Fnally, we show that locally nlotent dervatons on a Boolean semrng ossesng some condton must be zero. 2. Prelmnares In ths secton we recall the defnton of Boolean semrngs and some basc roertes of Boolean semrngs. Subrahmanyam, 1962 A Boolean semrng s a trle B, +, satsfyng the follow- Defnton 2.1 ng axoms: 1 B, + s an abelan grou, 2 B, s a semgrou, 3 a b + c =a b + a c for all a, b, c B, 4 a a = a for all a B, 5 a b c = b a c for all a, b, c n B. We denote a Boolean semrng B, +, by B wthout mentonng the oeratons, and denote a b by ab. Examle 2.2 Let B ={0, a, b, c}. Defne addton + and multlcaton as follows: + 0 a b c 0 0 a b c a a b c 0 b b c 0 a c c 0 a b 0 a b c 0 0 a b c a 0 a b c b 0 a b c c 0 a b c Page 2 of 7

3 Daowsud et al., Cogent Mathematcs 2017, 4: htts://do.org/ / Then B, +, s a Boolean semrng. Examle 2.3 Z 2, +, s a Boolean semrng where +, and are the addton, and multlcaton modulo 2, resectvely. The followng lemma summarses some basc roertes of Boolean semrng. The roof s straghtforward and hence omtted. Lemma 2.4 For any a, b, c n a Boolean semrng B, we have a =a, a0 = 0, a b = ab, v ab c =ab ac, v a + b = a b. Remark In Boolean semrng, 0a = 0, ab = ab, and a b =ab are not true n general, for nstance Examle 2.2, note that 0a = a, aa aa and a a aa. 3. Man results In what follows, let B denote a Boolean semrng unless otherwse secfed. Defnton 3.1 Let B be a Boolean semrng. A mang d:b B s called a dervaton on B f dx + y =dx+dy, and dxy =xdy+dxy, for all x, y B. Note that dxy = dxy + xdy holds for all x, y B because B, + s an abelean grou. Lemma 3.2 Let d be a dervaton on B. Then B satsfes the followng artal dstrbutve laws: xdy+ dxyz = xdyz + dxyz for all x, y, z B. Let x, y, z B. Then dxyz = xdyz + dxyz = xydz+ dyz+ dxyz = xydz + xdyz+dxyz, and dxyz =xydz+dxyz =xydz+xdy+dxyz. Comarng the two exressons above, we have xdy+dxyz =xdyz +dxyz. Ths comletes the roof. Note that dxyz = xdyz + dxyz. In the followng theorem, we obtan that the neccessary condton for the comoston of devaton on a Boolean semrng could be a dervaton. Page 3 of 7

4 Daowsud et al., Cogent Mathematcs 2017, 4: htts://do.org/ / Theorem 3.3 Let d 1 and d 2 be dervatons on B. Then the comoston of d 1 and d 2, d 1 s a dervaton on B f and only f d 1 xd 2 y+d 2 xd 1 y =0 for all x, y B. Suose that d 1 s a dervaton on B. Let x, y B. Then d 1 xy =xd 1 y+d 1 xy, and d 1 xy =d 1 xd 2 y+d 2 xy = xd 1 y+d 1 xd 2 y+d 2 xd 1 y+d 1 xy. By comarng the two exressons yelds d 1 xd 2 y+d 2 xd 1 y =0 for all x, y B. Conversely, assume that d 1 xd 2 y+d 2 xd 1 y =0 for all x, y B. Then d 1 xy =xd 1 y+d 1 xd 2 y+d 2 xd 1 y+d 1 xy = xd 1 y+d 1 xy, for all x, y B. Clearly, d 1 s an addtve mang on B. Thus d 1 s a dervaton on B. By Theorem 3.3, we obtan the followng corollary. Corollary 3.4 Let d 1 and d 2 be dervatons on B. If d 1 s a dervaton on B, so s d 2 d 1. Defnton 3.5 Let d be a dervaton on B. For any ostve nterger n, the nth dervaton of d s denoted by d n and s obtaned when d s comosed wth tself n tmes, and by d 0 x we mean the dentty functon defned by d 0 x =x. Next, we show that the nth dervaton satsfyng Lebnz rule. Lemma 3.6 Let d be a dervaton on B. For any ostve nteger n, d n xy = n n =0 d xd n y, for all x, y B. For n = 1, we have d 1 xy =xdy+dxy, for all x, y B. Let n > 1 and assume that the theorem holds for n 1. That s, n 1 n 1 d n 1 xy = =0 d xd n 1 y, for all x, y B. Then Page 4 of 7

5 Daowsud et al., Cogent Mathematcs 2017, 4: htts://do.org/ / n 1 n 1 d n xy =dd n 1 xy = d d xd n 1 y =0 n 1 n 1 = d xd n y+d +1 xd n 1 y =0 n 1 = xd n y+dxd n 1 y+ 1 n d 1 xd n +1 y+d xd n y 1 n 1 + d xd n y+d +1 xd n 1 y dxd n 1 y+d 2 xd n 2 y + + d n 1 xdy+d n xy [ ] n 1 n 1 = xd n y+ + + d xd n y+ + d n xy 1 n d n xy =xd n y+ + d xd n y+ + d n xy n n = d xd n y. =0 The roof s comleted. The characterstc of a Boolean semrng B s the smallest ostve nteger n such that a + a + + a n tmes = 0 for each a B. Corollary 3.7 Let B be a Boolean semrng of characterstc rme 2 and let d be a dervaton on B. Then d s a dervaton on B. Let x, y B. By Theorem 3.6, we obtan d xy = d xd y. Snce the characterstc of B s equal to and dvdes d xy = =0 0 xd y+ d xy. for all 1 k 1, we have Clearly, d s a addtve mang on B. Hence d s a dervaton on B. Defnton 3.8 A dervaton d on a Boolean semrng B s sad to be locally nlotent f for any b B, there exsts a ostve nteger n such that d n b =0. Defnton 3.9 A Boolean semrng B s called zero-symmtrc f 0a = 0 for all a B. Lemma 3.10 Let B be a zero-symmtrc Boolean semrng, x B, and d a dervaton on B. If d 2 x =0, then dx =0. Page 5 of 7

6 Daowsud et al., Cogent Mathematcs 2017, 4: htts://do.org/ / Assume that d 2 x =0. Then 0 = d 2 x =d 2 xx =xd 2 x+2dxdx+d 2 xx = 2dx, and by Lemma 3.2., dx =dxxdx =xdx+dxxdx =xdxdx +dxxdx = xdx + dx = 0. Therefore, dx =0. Lemma 3.11 Let B be a zero-symmtrc Boolean semrng, and let x B. If d s a dervaton on B satsfyng d n x =0 for some ostve nteger n 2, then d n 1 x =0. Assume that d n x =0 for some ostve nteger n 2. Then 0 = d n x =ddd n 2 xd n 2 x = d n 2 xd n x+d n 1 xd n 1 x+d n 1 xd n 1 x + d n xd n 2 x Hence d n 1 xd n 1 x+d n 1 xd n 1 x =0. We then have 0 = d n 1 xdd n 2 xd n 2 x + dd n 2 xd n 2 xd n 1 x = d n 1 xd n 2 xd n 1 x+d n 1 xd n 1 xd n 2 x + d n 2 xd n 1 xd n 1 x+d n 1 xd n 2 xd n 1 x = d n 2 xd n 1 xd n 1 x+d n 1 xd n 1 x + d n 1 xd n 2 x+d n 2 xd n 1 x. Hence d n 1 xd n 2 x+d n 2 xd n 1 x =0. It follows that d n 1 x =dd n 2 xd n 2 x = d n 1 xd n 2 x+d n 2 xd n 1 x =0 Therefore, d n 1 x =0. Theorem 3.12 Let B be a zero-symmtrc Boolean semrng. If d s a locally nlotent dervaton on B, then db =0. The roof s mmedately obtaned by Lemma Acknowledgements We are thankful to the anonymous referees for ther numerous helful suggestons. Fundng The authors receved no drect fundng for ths research. Author detals Katthaleeya Daowsud 1 E-mal: fsckyd@ku.ac.th Monrudee Srvoravt 1 E-mal: fscmdy@ku.ac.th Utsanee Leerawat 1 E-mal: fscutl@ku.ac.th ORCID ID: htt://orcd.org/ Faculty of Scence, Deartment of Mathematcs, Kasetsart Unversty, Bangkok 10900, Thaland. Ctaton nformaton Cte ths artcle as: On locally nlotent dervatons of Boolean semrngs, Katthaleeya Daowsud, Monrudee Srvoravt & Utsanee Leerawat, Cogent Mathematcs 2017, 4: References Ashraf, M., & Nadeem, R On dervatons and commutatvty n rme rngs. East-West Journal of Mathematcs, 3, Bell, H. E., & Argac, N Dervatons, roducts of dervatons and commutatvty n near-rngs. Algebra Colloquum, 8, Bresar, M A note on dervatons. Mathematcal Journal of Okayama Unversty, 32, Chebotar, M. A On a comoston of dervatons of rme rngs. Moscow Unversty Mathematcs Bulletn, 50, Page 6 of 7

7 Daowsud et al., Cogent Mathematcs 2017, 4: htts://do.org/ / Essen, A. v. d Locallynotent dervatons and ther alcatons. III, Journal of Pure and Aled Algebra, 98, Ferrero, M Y ves Lequan, and Andrzej Nowck, A note on locally notent devatons. Journal of Pure and Aled Algebra, North-Holland, 79, Galbat, J. L., & Verones, M. L On Boolean Semrngs. Isttuto Lombardo. Accadema d Scenze e Lettere. Rendcont, 114, Guzman, F The varety of Boolean semrngs. Journal of Pure and Aled Algebra, 78, Lee, P. H., & Lee, T. K Note on nlotent dervatons. Proceedngs of the Amercan Mathematcal Socety, 98, Posner, E. C Dervatons n rme rngs. Proceedngs of the Amercan Mathematcal Socety, 8, Subrahmanyam, N. V Boolean sem rngs. Mathematsche Annalen, 148, van den Essen, A., & Wang, X. K Dervatons n rme near-rngs. Proceedngs of the Amercan Mathematcal Socety, 121, The Authors. Ths oen access artcle s dstrbuted under a Creatve Commons Attrbuton CC-BY 4.0 lcense. You are free to: Share coy and redstrbute the materal n any medum or format Adat remx, transform, and buld uon the materal for any urose, even commercally. The lcensor cannot revoke these freedoms as long as you follow the lcense terms. Under the followng terms: Attrbuton You must gve arorate credt, rovde a lnk to the lcense, and ndcate f changes were made. You may do so n any reasonable manner, but not n any way that suggests the lcensor endorses you or your use. No addtonal restrctons You may not aly legal terms or technologcal measures that legally restrct others from dong anythng the lcense ermts. Cogent Mathematcs ISSN: s ublshed by Cogent OA, art of Taylor & Francs Grou. Publshng wth Cogent OA ensures: Immedate, unversal access to your artcle on ublcaton Hgh vsblty and dscoverablty va the Cogent OA webste as well as Taylor & Francs Onlne Download and ctaton statstcs for your artcle Rad onlne ublcaton Inut from, and dalog wth, exert edtors and edtoral boards Retenton of full coyrght of your artcle Guaranteed legacy reservaton of your artcle Dscounts and wavers for authors n develong regons Submt your manuscrt to a Cogent OA journal at Page 7 of 7