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A note on multiplicative (generalized) derivations with annihilator conditions

  • Basudeb Dhara EMAIL logo and Krishna Gopal Pradhan

Abstract

Let R be a prime ring with center Z(R), aR (a ≠ 0) and I a nonzero ideal of R. Suppose that F,d:RR are any two mappings such that F(xy)=F(x)y+xd(y) for all x,yR. For all x,yI, we investigate the identities a(F(xy)±xy)=0, a(F(xy)±yx)=0, a(F(x)F(y)±xy)=0, a(F(x)F(y)±yx)=0, a(d(x)F(y)±xy)Z(R), a(d(x)F(y)±yx)Z(R) and a(F(xy)±F(x)F(y))=0.

The authors would like to thank the referee for providing very helpful comments and suggestions.

References

1 S. Ali, B. Dhara and A. Fošner, Some commutativity theorems concerning additive mappings and derivations on semiprime rings, Contemporary Ring Theory 2011, World Scientific, Hackensack (2012), 135–143. 10.1142/9789814397681_0012Search in Google Scholar

2 M. Ashraf, A. Ali and S. Ali, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math. 31 (2007), 3, 415–421. Search in Google Scholar

3 M. Ashraf, A. Ali and R. Rani, On generalized derivations of prime rings, Southeast Asian Bull. Math. 29 (2005), 4, 669–675. Search in Google Scholar

4 M. Ashraf and N. Rehman, On derivation and commutativity in prime rings, East-West J. Math. 3 (2001), 1, 87–91. 10.1007/BF03323547Search in Google Scholar

5 K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with Generalized Identities, Pure Appl. Math. 196, Dekker, New York, 1996. Search in Google Scholar

6 H. E. Bell, W. S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1987), 1, 92–101. 10.4153/CMB-1987-014-xSearch in Google Scholar

7 M. Brešar, On the distance of the composition of two derivations to the generalized derivations, Glasg. Math. J. 33 (1991), 1, 89–93. 10.1017/S0017089500008077Search in Google Scholar

8 C. M. Chang and Y. C. Lin, Derivations on one-sided ideals of prime rings, Tamsui Oxf. J. Inf. Math. Sci. 17 (2001), 2, 139–145. Search in Google Scholar

9 M. N. Daif, When is a multiplicative derivation additive?, Int. J. Math. Math. Sci. 14 (1991), 3, 615–618. 10.1155/S0161171291000844Search in Google Scholar

10 M. N. Daif, Commutativity results for semiprime rings with derivations, Int. J. Math. Math. Sci. 21 (1998), 3, 471–474. 10.1155/S0161171298000660Search in Google Scholar

11 M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci. 15 (1992), 1, 205–206. 10.1155/S0161171292000255Search in Google Scholar

12 M. N. Daif and M. S. Tammam El-Sayiad, Multiplicative generalized derivations which are additive, East-West J. Math. 9 (2007), 1, 31–37. 10.12988/ija.2007.07060Search in Google Scholar

13 B. Dhara, Remarks on generalized derivations in prime and semiprime rings, Int. J. Math. Math. Sci. 2010 (2010), Article ID 646587. 10.1155/2010/646587Search in Google Scholar

14 B. Dhara and S. Ali, On multiplicative (generalized)-derivations in prime and semiprime rings, Aequationes Math. 86 (2013), 1–2, 65–79. 10.1007/s00010-013-0205-ySearch in Google Scholar

15 H. Goldmann and P. Šemrl, Multiplicative derivations on C(X), Monatsh. Math. 121 (1996), 3, 189–197. 10.1007/BF01298949Search in Google Scholar

16 I. N. Herstein, Rings with Involution, Chicago Lectures in Math., The University of Chicago Press, Chicago, 1976. Search in Google Scholar

17 B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), 4, 1147–1166. 10.1080/00927879808826190Search in Google Scholar

18 W. S. Martindale III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21 (1969), 695–698. 10.1090/S0002-9939-1969-0240129-7Search in Google Scholar

19 M. Marubayashi, M. Ashraf, N. Rehman and S. Ali, On generalized (α,β)-derivations in prime rings, Algebra Colloq. 17 (2010), 1, 865–874. 10.1142/S1005386710000805Search in Google Scholar

20 E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. 10.1090/S0002-9939-1957-0095863-0Search in Google Scholar

21 M. A. Quadri, M. S. Khan and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34 (2003), 9, 1393–1396. Search in Google Scholar

22 N. Rehman, On commutativity of rings with generalized derivations, Math. J. Okayama Univ. 44 (2002), 43–49. 10.1016/j.joems.2014.12.011Search in Google Scholar

Received: 2013-8-13
Revised: 2015-11-21
Accepted: 2016-2-26
Published Online: 2016-4-27
Published in Print: 2016-6-1

© 2016 by De Gruyter

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