Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings

Vincenzo de Filippis

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 271-292
  • ISSN: 0011-4642

Abstract

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Let R be a prime ring of characteristic different from 2, Q r its right Martindale quotient ring and C its extended centroid. Suppose that F , G are generalized skew derivations of R with the same associated automorphism α , and p ( x 1 , ... , x n ) is a non-central polynomial over C such that [ F ( x ) , α ( y ) ] = G ( [ x , y ] ) for all x , y ∈ { p ( r 1 , ... , r n ) : r 1 , ... , r n ∈ R } . Then there exists λ ∈ C such that F ( x ) = G ( x ) = λ α ( x ) for all x ∈ R .

How to cite

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de Filippis, Vincenzo. "Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings." Czechoslovak Mathematical Journal 66.1 (2016): 271-292. <http://eudml.org/doc/276784>.

@article{deFilippis2016,
abstract = {Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha $, and $p(x_1,\ldots ,x_n)$ is a non-central polynomial over $C$ such that \[ [F(x),\alpha (y)]=G([x,y]) \] for all $x,y \in \lbrace p(r_1,\ldots ,r_n)\colon r_1,\ldots ,r_n \in R\rbrace $. Then there exists $\lambda \in C$ such that $F(x)=G(x)=\lambda \alpha (x)$ for all $x\in R$.},
author = {de Filippis, Vincenzo},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized skew derivation; prime ring},
language = {eng},
number = {1},
pages = {271-292},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings},
url = {http://eudml.org/doc/276784},
volume = {66},
year = {2016},
}

TY - JOUR
AU - de Filippis, Vincenzo
TI - Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 271
EP - 292
AB - Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha $, and $p(x_1,\ldots ,x_n)$ is a non-central polynomial over $C$ such that \[ [F(x),\alpha (y)]=G([x,y]) \] for all $x,y \in \lbrace p(r_1,\ldots ,r_n)\colon r_1,\ldots ,r_n \in R\rbrace $. Then there exists $\lambda \in C$ such that $F(x)=G(x)=\lambda \alpha (x)$ for all $x\in R$.
LA - eng
KW - generalized skew derivation; prime ring
UR - http://eudml.org/doc/276784
ER -

References

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  1. Arga{ç}, N., Carini, L., Filippis, V. De, 10.11650/twjm/1500574164, Taiwanese J. Math. 12 (2008), 419-433. (2008) Zbl1153.16029MR2402125DOI10.11650/twjm/1500574164
  2. Brešar, M., Miers, C. R., 10.4153/CMB-1994-066-4, Can. Math. Bull. 37 (1994), 457-460. (1994) MR1303671DOI10.4153/CMB-1994-066-4
  3. Chang, J.-C., 10.11650/twjm/1500407520, Taiwanese J. Math. 7 (2003), 103-113. (2003) Zbl1048.16018MR1961042DOI10.11650/twjm/1500407520
  4. Chuang, C.-L., 10.1006/jabr.1999.8052, J. Algebra 224 (2000), 292-335. (2000) MR1739582DOI10.1006/jabr.1999.8052
  5. Chuang, C.-L., 10.1006/jabr.1993.1181, J. Algebra 160 (1993), 130-171. (1993) MR1237081DOI10.1006/jabr.1993.1181
  6. Chuang, C.-L., 10.1016/0021-8693(92)90023-F, J. Algebra 149 (1992), 371-404. (1992) MR1172436DOI10.1016/0021-8693(92)90023-F
  7. Chuang, C.-L., 10.1090/S0002-9939-1988-0947646-4, Proc. Am. Math. Soc. 103 (1988), 723-728. (1988) Zbl0656.16006MR0947646DOI10.1090/S0002-9939-1988-0947646-4
  8. Chuang, C.-L., 10.1007/BF02779669, Isr. J. Math. 59 (1987), 98-106. (1987) MR0923664DOI10.1007/BF02779669
  9. Chuang, C.-L., Lee, T.-K., 10.1016/j.jalgebra.2003.12.032, J. Algebra 288 (2005), 59-77. (2005) Zbl1073.16021MR2138371DOI10.1016/j.jalgebra.2003.12.032
  10. Chuang, C.-L., Lee, T.-K., Rings with annihilator conditions on multilinear polynomials, Chin. J. Math. 24 (1996), 177-185. (1996) MR1401645
  11. Filippis, V. De, 10.1007/BF03191235, Collect. Math. 61 (2010), 303-322. (2010) Zbl1232.16029MR2732374DOI10.1007/BF03191235
  12. Vincenzo, O. M. Di, On the n th centralizer of a Lie ideal, Boll. Unione Mat. Ital., VII. Ser. 3-A (1989), 77-85. (1989) MR0990089
  13. Faith, C., Utumi, Y., 10.1007/BF01895723, Acta Math. Acad. Sci. Hung. 14 (1963), 369-371. (1963) Zbl0147.28602MR0155858DOI10.1007/BF01895723
  14. Herstein, I. N., Topics in Ring Theory, Chicago Lectures in Mathematics The University of Chicago Press, Chicago (1969). (1969) Zbl0232.16001MR0271135
  15. Jacobson, N., 10.1007/BFb0070023, Lecture Notes in Mathematics 441 Springer, Berlin (1975). (1975) MR0369421DOI10.1007/BFb0070023
  16. Jacobson, N., 10.1090/coll/037, American Mathematical Society Colloquium Publications 37 AMS, Providence (1956). (1956) Zbl0073.02002MR0081264DOI10.1090/coll/037
  17. Lanski, C., Montgomery, S., 10.2140/pjm.1972.42.117, Pac. J. Math. 42 (1972), 117-136. (1972) MR0323839DOI10.2140/pjm.1972.42.117
  18. Lin, J.-S., Liu, C.-K., Strong commutativity preserving maps on Lie ideals, Linear Algebra Appl. 428 (2008), 1601-1609. (2008) Zbl1141.16021MR2388643
  19. Liu, C.-K., 10.1007/s00605-010-0281-1, Monatsh. Math. 166 (2012), 453-465. (2012) Zbl1250.16029MR2925149DOI10.1007/s00605-010-0281-1
  20. Liu, C.-K., Liau, P.-K., 10.1080/03081087.2010.535819, Linear Multilinear Algebra 59 (2011), 905-915. (2011) Zbl1230.16033MR2826060DOI10.1080/03081087.2010.535819
  21. Ma, J., Xu, X. W., Niu, F. W., 10.1007/s10114-008-7445-0, Acta Math. Sin., Engl. Ser. 24 (2008), 1835-1842. (2008) Zbl1169.16020MR2453063DOI10.1007/s10114-008-7445-0
  22. III, W. S. Martindale, 10.1016/0021-8693(69)90029-5, J. Algebra 12 (1969), 576-584. (1969) MR0238897DOI10.1016/0021-8693(69)90029-5
  23. Posner, E. C., 10.1090/S0002-9939-1957-0095863-0, Proc. Am. Math. Soc. 8 (1958), 1093-1100. (1958) Zbl0082.03003MR0095863DOI10.1090/S0002-9939-1957-0095863-0
  24. Wong, T.-L., Derivations with power-central values on multilinear polynomials, Algebra Colloq. 3 (1996), 369-378. (1996) Zbl0864.16031MR1422975

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