Vietnam Journal of Mathematics

, Volume 43, Issue 4, pp 819–828

# On Additive Mappings in a ∗-Ring with an Identity Element

• Abu Zaid Ansari
Article

## Abstract

Let R be a semiprime ring with involution ∗ and let F, D : RR be additive mappings satisfying the conditions (i) F(x 2) = F(x)x +x D(x) and D(x 2) = D(x)x +x D(x); (ii) F(x n + 1) = F(x)(x ) n +x D(x)(x ) n − 1+(x )2 D(x)(x ) n − 2+⋯+(x ) n D(x) for all xR. Then, F(x y) = F(y)x +y D(x) and D(x y) = D(y)x +y D(x) for all x, yR.

## Keywords

Additive mappings Semiprime rings and involution

## Mathematics Subject Classification (2010)

16W25 16N60 16R50

## Notes

### Acknowledgments

The authors are greatly indebted to the referee for his/her several useful suggestions and valuable comments to improve the presentation of this paper.

## References

1. 1.
Ashraf, M., Rehman, N.: On Jordan generalized derivations in rings. Math. J. Okayama Univ. 42, 7–10 (2000)
2. 2.
Brešar, M.: Jordan mappings of semiprime rings. J. Algebra 127, 218–228 (1989)
3. 3.
Bresar, M.: On the distance of the composition of the two derivations to the generalized derivations. Glasg. Math. J. 33, 89–93 (1991)
4. 4.
Bresar, M., Vukman, J.: Jordan derivation of prime rings. Bull. Aust. Math. Soc. 37, 321–322 (1988)
5. 5.
Cusack, J.M.: Jordan derivations in rings. Proc. Am. Math. Soc. 53, 321–324 (1975)
6. 6.
Dhara, B., Sharma, R.K.: On additive mappings in rings with identity element. Int. Math. Forum 4, 727–732 (2009)
7. 7.
Herstein, I.N.: Jordan derivations of prime rings. Proc. Am. Math. Soc. 8, 1104–1110 (1957)
8. 8.
Herstein, I.N.: Topics in Ring Theory. Univ. Chicago Press, Chicago (1969)Google Scholar
9. 9.
Jing, W., Lu, S.: Generalized Jordan derivations on prime rings and standard operator algebras. Taiwan. J. Math. 7, 605–613 (2003)
10. 10.
Kosi-Ulbl, I.: A remark on centralizers in semiprime rings. Glas. Math. 39, 21–26 (2004)
11. 11.
Lanski, C.: An Engel condition with derivation for left ideals. Proc. Am. Math. Soc. 125, 339–345 (1997)
12. 12.
Rehman, N., Ansari, A.Z.: Additive mappings of semiprime rings with involution. Aligarh Bull. Math. 30, 117–123 (2011)Google Scholar
13. 13.
Rehman, N., Ansari, A.Z., Bano, T.: On generalized Jordan ∗-derivation in rings. J. Egypt. Math. Soc. 22, 11–13 (2014)
14. 14.
Vukman, J.: A note on generalized derivations of semiprime rings. Taiwan. J. Math. 11, 367–370 (2007)
15. 15.
Vukman, J., Kosi-Ulbl, I.: An equation related to centralizers in semiprime rings. Glas. Math. 38, 253–261 (2003)
16. 16.
Zalar, B.: On centralizers of semiprime rings. Comment. Math. Univ. Carol. 32, 609–614 (1991)