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Let $R$ be a ring with characteristic different from 2. An additive mapping $F:R\rightarrow R$ is called a generalized derivation on $R$ if there exists a derivation $d:R\rightarrow R$ such that $F(xy)=F(x)y+xd(y)$ holds for all $x,y\in R$. In the present paper, we show that if $R$ is a prime ring satisfying certain identities involving a generalized derivation $F$ associated with a derivation $d$, then $R$ becomes commutative and in some cases $d$ comes out to be zero (i.e., $F$ becomes a left centralizer). We provide some counter examples to justify that the restrictions imposed in the hypotheses of our theorems are not superfluous.
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참고문헌 (13)
참고문헌 신청
Mohammad Khalil Abu Nawas / 2018 / On Ideals and Commutativity of Prime Rings with Generalized Derivations / European Journal of Pure and Applied Mathematics 11(1):79~89
M. Ashraf / 2007 / Some commutativity theorems for rings with generalized derivations / Southeast Asian Bull. Math 31(3):415~421
Mohammad Ashraf / 2018 / Generalized Derivations on *-prime Rings / Kyungpook Mathematical Journal 58(3):481~488
M. Ashraf / 2001 / On derivation and commutativity in prime rings / East-West J. Math 3(1):87~91
H. E. Bell / 1995 / On derivations and commutativity in prime rings / Acta Math. Hungar 66(4):337~343
M. Ashraf / 2007 / Some commutativity theorems for rings with generalized derivations / Southeast Asian Bull. Math 31(3):415~421
Mohammad Ashraf / 2018 / Generalized Derivations on *-prime Rings / Kyungpook Mathematical Journal 58(3):481~488
M. Ashraf / 2001 / On derivation and commutativity in prime rings / East-West J. Math 3(1):87~91
H. E. Bell / 1995 / On derivations and commutativity in prime rings / Acta Math. Hungar 66(4):337~343
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