$b$-generalized derivations on multilinear polynomials in prime rings
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Let $R$ be a noncommutative prime ring of characteristic different from $2$, $Q$ be its maximal right ring of quotients and $C$ be its extended centroid. Suppose that $f(x_1,\ldots,x_n)$ be a noncentral multilinear polynomial over $C$, $b\in Q$, $F$ a $b$-generalized derivation of $R$ and $d$ is a nonzero derivation of $R$ such that $$d([F(f(r)),f(r)])=0$$ for all $r=(r_1,\ldots,r_n)\in R^n$. Then one of the following holds: (1) there exists $\lambda\in C$ such that $F(x)=\lambda x$ for all $x\in R$; (2) there exist $\lambda\in C$ and $p\in Q$ such that $F(x)=\lambda x+px+xp$ for all $x\in R$ with $f(x_1,\ldots,x_n)^2$ is central valued in $R$.
Abstract
Let $R$ be a noncommutative prime ring of characteristic different from $2$, $Q$ be its maximal right ring of quotients and $C$ be its extended centroid. Suppose that $f(x_1,\ldots,x_n)$ be a noncentral multilinear polynomial over $C$, $b\in Q$, $F$ a $b$-generalized derivation of $R$ and $d$ is a nonzero derivation of $R$ such that $$d([F(f(r)),f(r)])=0$$ for all $r=(r_1,\ldots,r_n)\in R^n$. Then one of the following holds: (1) there exists $\lambda\in C$ such that $F(x)=\lambda x$ for all $x\in R$; (2) there exist $\lambda\in C$ and $p\in Q$ such that $F(x)=\lambda x+px+xp$ for all $x\in R$ with $f(x_1,\ldots,x_n)^2$ is central valued in $R$.
참고문헌 (20)
- K. I. Beidar, "Rings with Generalized Identities, Pure Appl. Math., 196" (1996), Marcel Dekker
- C. L. Chuang, "GPIs having coeffcients in Utumi quotient rings", Proc. Amer. Math. Soc., 103(3), pp.723~728 (1988)[DOI]
- C. L. Chuang, "Derivations modulo elementary operators", J. Algebra, 338, pp.56~70 (2011)[DOI]
- V. De Filippis, "An Engel condition with generalized derivations on multilinear polynomials", Israel J. Math., 162, pp.93~108 (2007)[DOI]
- V. De Filippis, "Posner's second theorem, multilinear polynomials and vanishing derivations", J. Aust. Math. Soc., 76(3), pp.357~368 (2004)[DOI]
- V. De Filippis, "Vanishing derivations and centralizers of generalized derivations on multilinear polynomials", Comm. Algebra, 40(6), pp.1918~1932 (2012)[DOI]
- C. Demir, "Prime rings with generalized derivations on right ideals", Algebra Colloq., 18(1), pp.987~998 (2011)[DOI]
- B. Dhara, "Generalized derivations acting on multilinear polynomials in prime rings and Banach algebras", Commun. Math. Stat., 4(1), pp.39~54 (2016)[DOI]
- N. Jacobson, "Structure of rings, Amer. Math. Soc. Colloq. Pub., 37" (1964)
- V. K. Kharchenko, "Differential identity of prime rings", Algebra i Logika, 17(2), pp.220~ (1978)
- M. T. Kosan, "b-Generalized derivations having nilpotent values", J. Aust. Math. Soc., 96(3), pp.326~337 (2014)[DOI]
- C. Lanski, "Differential identities, Lie ideals, and Posner's theorems", Pacific J. Math., 134(2), pp.275~297 (1988)[DOI]
- P. H. Lee, "Derivations with Engel conditions on multilinear polynomials", Proc. Amer. Math. Soc., 124(9), pp.2625~2629 (1996)[DOI]
- T. K. Lee, "Semiprime rings with differential identities", Bull. Inst. Math. Acad. Sinica, 20(1), pp.27~38 (1992)
- T. K. Lee, "Additive maps having a generalized derivation expansion", J. Algebra Appl., 14(4), pp.13~ (2015)
- U. Leron, "Nil and power central polynomials in rings", Trans. Amer. Math. Soc., 202, pp.97~103 (1975)[DOI]
- C.-K. Liu, "An Engel condition with b-generalized derivations", Linear Multilinear Algebra, 65(2), pp.300~312 (2017)[DOI]
- W. S. Martindale III, "Prime rings satisfying a generalized polynomial identity", J. Algebra, 12, pp.576~584 (1969)[DOI]
- E. C. Posner, "Derivations in prime rings", Proc. Amer. Math. Soc., 8, pp.1093~1100 (1957)[DOI]
- T. L. Wong, "Derivations with power central values on multilinear polynomials", Algebra Colloq., 3(4), pp.369~478 (1996)