Smash products and differential identities
Authors:
ChenLian Chuang and YuanTsung Tsai
Journal:
Trans. Amer. Math. Soc. 364 (2012), 41554168
MSC (2010):
Primary 16S40, 16S32, 16W25, 16S36, 16S30
Published electronically:
March 21, 2012
MathSciNet review:
2912449
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be the universal enveloping algebra of a Lie algebra and a module algebra, where is considered as a Hopf algebra canonically. We determine the centralizer of in with its associated graded algebra. We then apply this to the Ore extension , where . With the help of PBWbases, the following is proved for a prime ring : Let be the symmetric Martindale quotient ring of . For , for all iff , where is over the centralizer of in . Finally, we deduce from this Kharchenko's theorem on differential identities.
 1.
S.
A. Amitsur, Derivations in simple rings, Proc. London Math.
Soc. (3) 7 (1957), 87–112. MR 0088480
(19,525d)
 2.
K.
I. Beidar, W.
S. Martindale III, and A.
V. Mikhalev, Rings with generalized identities, Monographs and
Textbooks in Pure and Applied Mathematics, vol. 196, Marcel Dekker,
Inc., New York, 1996. MR 1368853
(97g:16035)
 3.
V.
D. Burkov, Differentially prime rings, Uspekhi Mat. Nauk
35 (1980), no. 5(215), 219–220 (Russian). MR 595145
(82f:16002)
 4.
ChenLian
Chuang and YuanTsung
Tsai, Higher derivations of Ore extensions by 𝑞skew
derivations, J. Pure Appl. Algebra 214 (2010),
no. 10, 1778–1786. MR 2608105
(2011e:16050), 10.1016/j.jpaa.2009.12.020
 5.
ChenLian
Chuang, TsiuKwen
Lee, ChengKai
Liu, and YuanTsung
Tsai, Higher derivations of Ore extensions, Israel J. Math.
175 (2010), 157–178. MR 2607542
(2011i:16072), 10.1007/s118560100007z
 6.
V.
K. Harčenko, Differential identities of prime rings,
Algebra i Logika 17 (1978), no. 2, 220–238,
242–243 (Russian). MR 541758
(81f:16025)
 7.
V.
K. Harčenko, Differential identities of semiprime
rings, Algebra i Logika 18 (1979), no. 1,
86–119, 123 (Russian). MR 566776
(81f:16052)
 8.
André
Leroy and Jerzy
Matczuk, The extended centroid and 𝑋inner automorphisms of
Ore extensions, J. Algebra 145 (1992), no. 1,
143–177. MR 1144664
(93b:16053), 10.1016/00218693(92)90182L
 9.
Jerzy
Matczuk, Extended centroids of skew polynomial rings, Math. J.
Okayama Univ. 30 (1988), 13–20. MR 976726
(89m:16006)
 10.
Susan
Montgomery, Hopf algebras and their actions on rings, CBMS
Regional Conference Series in Mathematics, vol. 82, Published for the
Conference Board of the Mathematical Sciences, Washington, DC; by the
American Mathematical Society, Providence, RI, 1993. MR 1243637
(94i:16019)
 11.
Christophe
Reutenauer, Free Lie algebras, London Mathematical Society
Monographs. New Series, vol. 7, The Clarendon Press, Oxford University
Press, New York, 1993. Oxford Science Publications. MR 1231799
(94j:17002)
 12.
JeanPierre
Serre, Lie algebras and Lie groups, 2nd ed., Lecture Notes in
Mathematics, vol. 1500, SpringerVerlag, Berlin, 1992. 1964 lectures
given at Harvard University. MR 1176100
(93h:17001)
 13.
Moss
E. Sweedler, Hopf algebras, Mathematics Lecture Note Series,
W. A. Benjamin, Inc., New York, 1969. MR 0252485
(40 #5705)
 14.
YuanTsung
Tsai and ChenLian
Chuang, Quotient rings of Ore extensions with more than one
indeterminate, Comm. Algebra 36 (2008), no. 10,
3608–3615. MR 2458396
(2009k:16054), 10.1080/00927870802157749
 15.
YuanTsung
Tsai, TsuYang
Wu, and ChenLian
Chuang, Jacobson radicals of Ore extensions of derivation
type, Comm. Algebra 35 (2007), no. 3,
975–982. MR 2305244
(2007m:16045), 10.1080/00927870601117613
 1.
 S. A. Amitsur, Derivations in simple rings, Proc. London Math. Soc. (3) (1957), 87112. MR 0088480 (19:525d)
 2.
 K. I. Beidar, W. S. Martindale, III, A. V. Mikhalev, ``Rings with generalized identities'', Monographs and Textbooks in Pure and Applied Mathematics , Marcel Dekker, Inc., New York, 1996. MR 1368853 (97g:16035)
 3.
 V. D. Burkov, On differentially prime rings, (Russian) Uspekhi Mat. Nauk (5) (1980), 219220. (Engl. Transl. Russian Math. Surveys 35(5):253254.) MR 595145 (82f:16002)
 4.
 C.L. Chuang and Y.T. Tsai, Higher derivations of Ore extensions by skew derivations, Journal of Pure and Applied Algebra, (10) (2010), 17781786. MR 2608105
 5.
 C.L. Chuang, T.K. Lee, C.K. Liu and Y.T. Tsai, Higher Derivations of Ore Extensions, Israel J. Math (2010), 157178. MR 2607542
 6.
 V. K. Kharchenko, Differential identities of prime rings, (Russian) Algebra i Logika (2) (1978), 220238. (Engl. Transl., Algebra and Logic (2) (1978), 154168.) MR 541758 (81f:16025)
 7.
 V. K. Kharchenko, Differential identities of semiprime rings, (Russian) Algebra i Logika (1) (1979), 86119. (Engl. Transl., Algebra and Logic (1) (1979), 5880.) MR 566776 (81f:16052)
 8.
 A. Leroy and J. Matczuk, The extended centroid and inner automorphisms of Ore extensions, J. Algebra (1) (1992), 143177. MR 1144664 (93b:16053)
 9.
 J. Matczuk, Extended centroids of skew polynomial rings, Math. J. Okayama Univ. (1988), 1320. MR 976726 (89m:16006)
 10.
 S. Montgomery, ``Hopf algebras and their actions on rings'', Regional conference series in mathematics; no. 82, American Mathematical Society, Providence, Rhode Island, 1992. MR 1243637 (94i:16019)
 11.
 C. Reutenauer, ``Free Lie algebras'', London Mathematical Society monographs; new ser. 7, Oxford: Clarendon Press; New York, Oxford University Press, 1993. MR 1231799 (94j:17002)
 12.
 J.P. Serre, ``Lie algebras and Lie groups: 1964 lectures given at Harvard University'', (1992), 2nd ed., Lecture Notes in Mathematics, 1500, SpringerVerlag: Berlin. MR 1176100 (93h:17001)
 13.
 M. E. Sweedler, ``Hopf Algebras'', Mathematics Lecture Notes Series, 1969, W. A. Benjamin, Inc., New York, 1996. MR 0252485 (40:5705)
 14.
 Y.T. Tsai and C.L. Chuang, Quotient Rings of Ore Extensions with More Than One Indeterminate, Commun. Algebra (10), (2008), 36083615. MR 2458396 (2009k:16054)
 15.
 Y.T. Tsai, T.Y. Wu, and C.L. Chuang, Jacobson radicals of Ore extensions of derivation type, Commun. Algebra (3) (2007), 975982. MR 2305244 (2007m:16045)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
16S40,
16S32,
16W25,
16S36,
16S30
Retrieve articles in all journals
with MSC (2010):
16S40,
16S32,
16W25,
16S36,
16S30
Additional Information
ChenLian Chuang
Affiliation:
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
Email:
chuang@math.ntu.edu.tw
YuanTsung Tsai
Affiliation:
Department of Applied Mathematics, Tatung University, Taipei 104, Taiwan
Email:
yttsai@ttu.edu.tw
DOI:
http://dx.doi.org/10.1090/S000299472012054547
Keywords:
Derivations,
universal enveloping algebras,
centralizers,
smash products,
Ore extensions,
differential identities
Received by editor(s):
May 4, 2010
Received by editor(s) in revised form:
August 30, 2010
Published electronically:
March 21, 2012
Dedicated:
To PjekHwee Lee on his retirement
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
