Annihilator conditions with generalized skew derivations and Lie ideals of prime rings
Let $R$ be a prime ring, $Q_r$ its right Martindale quotient ring, $L$ a non-central Lie ideal of $R$, $n\geq 1$ a fixed integer, $F$ and $G$ two generalized skew derivations of $R$ with the same associated automorphism, $p\in R$ a fixed element. If $p\bigl(F(x)F(y)-G(y)x\bigr)^n=0$, for any $x,y \in L$, then there exist $a,c\in Q_r$ such that $F(x)=ax$ and $G(x)=cx$, for any $x\in R$, with $pa=pc=0$, unless when $R$ satisfies the standard polynomial identity $s_4(x_1,\ldots,x_4)$.
Keywords:
Generalized, skew derivation, prime ring,
___
- J.-C. Chang, Annihilators of power values of a right generalized $(\alpha,\beta)$-derivation, Bull. Inst. Math. Acad. Sin., 4 (2009), 67-73.
- J.-C. Chang, Generalized skew derivations with nilpotent values on Lie ideals, Monatsh. Math., 161 (2010), 155-160.
- C.-L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103 (1988), 723-728.
- C.-L. Chuang, Differential identities with automorphisms and antiautomorphisms I, J. Algebra, 149 (1992), 371-404.
- C.-L. Chuang, Differential identities with automorphisms and antiautomorphisms II, J. Algebra, 160 (1993), 130-171.
- C.-L. Chuang, Identities with skew derivations, J. Algebra, 224 (2000), 292-335.
- C.-L. Chuang, M.-C. Chou and C.-K. Liu, Skew derivations with annihilating Engel conditions, Publ. Math. Debrecen, 68 (2006), 161-170.
- V. De Filippis, Annihilators of power values of generalized derivations on multilinear polynomials, Bull. Aust. Math. Soc., 80 (2009), 217-232.
- V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra, 40 (2012), 1918-1932.
- V. De Filippis and O. M. Di Vincenzo, Generalized skew derivations and nilpotent values on Lie ideals, Algebra Colloq., 26 (2019), 589-614.
- V. De Filippis, N. Rehman and G. Scudo, Certain functional identities involving a pair of generalized skew derivations with nilpotent values on Lie ideals, pre-print.
- O. M. Di Vincenzo, On the n-th centralizer of a Lie ideal, Boll. Un. Mat. Ital. A (7), 3 (1989), 77-85.
- T. S. Erickson, W. S. Martindale III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60 (1975), 49-63.
- M. Eroğlu and N. Argaç, Power values of generalized skew derivations with annihilator conditions on Lie ideals, Bull. Iranian Math. Soc., 46 (2020), 1583-1598.
- I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, 1969.
- N. Jacobson, Structure of Rings, Amer. Math. Soc., Providence, RI, 1964.
- C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math., 42 (1) (1972), 117-136.
- W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584.
- R. K. Sharma, B. Dhara, V. De Filippis and C. Garg, A result concerning nilpotent values with generalized skew derivations on Lie ideals, Comm. Algebra, 46 (2018), 5330-5341.
- N. Yarbil and N. Argaç, Annihilators of power values of generalized skew derivations on Lie ideals, Algebra and its applications, 307-316, De Gruyter Proc. Math., De Gruyter, Berlin, 2018.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI