A note on generalized derivations on prime rings
Tóm tắt
Let R be a prime ring with the extended centroid C and symmetric Martindale quotient ring
$$Q_s(R)$$
. In this paper we prove the following result. Let
$$F: R \rightarrow R$$
be a generalized derivation associated with a non-zero derivation d on R and let h be an additive map of R such that
$$F(x)x=xh(x)$$
for all
$$x\in R$$
. Then either R is commutative or
$$F(x)=xp$$
and
$$h(x)=px$$
where
$$p\in Q_{s}(R)$$
.
Tài liệu tham khảo
Ashraf, M.; Rehman, N.; Ali, S.; Mozumder, M.R.: On semiprime rings with generalized derivations. Bol. Soc. Parana de Mat. 28, 15–22 (2010)
Beidar, K.I.; Brešar, M.; Chebotar, M.A.: Functional identities revisited: the functional and the strong degree. Commun. Algebra 30, 935–969 (2002)
Beidar, K.I.; Martindale III, W.S.; Mikhalev, A.V.: Rings with Generalized Identities, Monographs and Textbooks in Pure and Applied Mathematics, vol. 196. Marcel Dekker, Inc., New York (1996)
Brešar, M.: On the distance of the composition of two derivations to the generalized derivations. Glasgow Math. J. 33, 89–93 (1991)
Brešar, M.: Centralizing mappings and derivations in prime rings. J. Algebra 156, 385–394 (1993)
Brešar, M.: On skew-commuting mappings of rings. Bull. Aust. Math. Soc. 47, 291–296 (1993)
Brešar, M.: Functional identities of degree two. J. Algebra 172, 690–720 (1995)
Herstein, I.N.: Topics in Ring Theory. University of Chicago Press, Chicago (1969)
Hvala, B.: Generalized derivations in rings. Commun. Algebra 26, 1147–1166 (1998)
Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sinica 20(1), 27–38 (1992)
Lee, T.K.: Generalized derivations of left faithful rings. Commun. Algebra 27(8), 4057–4073 (1999)
Lanski, C.: Differental identities, Lie ideals, and Posner’s theorems. Pac. J. Math. 134, 275–297 (1988)
Lanski, C.: Lie ideals and central identities with derivation. Can. J. Math. 44, 553–560 (1992)
Martindale III, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969)
Posner, E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8, 1093–1100 (1957)