A Characterization of Semiprime Rings with Homoderivations

Abstract

This paper is focused on the commutativity of the laws of semiprime rings, which satisfy some algebraic identities involving homoderivations on ideals. It provides new and notable results that will interest researchers in this field, such as “R contains a nonzero central ideal if R admits a nonzero homoderivation δ on I such that δ(I)⊆Z where R is a semiprime ring with center Z and I a nonzero ideal of R”. Moreover, the research also generalizes some results previously published in the literature, including derivation on prime rings using homoderivation semiprime rings. It also demonstrates the necessity of hypotheses operationalized in theorems by an example. Finally, the paper discusses how the results herein can be further developed in future research.

Keywords

• Semiprime rings
• ideals
• derivations
• homoderivations

References

1. H. E. Bell, M. N. Daif, On Commutativity and Strong Commutativity-Preserving Mappings, Canadian Mathematical Bulletin 37 (1994) 443–447.
2. M. Bresar, Commuting Traces of Biadditive Mappings, Commutativity Preserving Mappings and Lie Mappings, Transactions of the American Mathematical Society 335 (2) (2003) 525–546.
3. J. Ma, X. W. Xu, Strong Commutativity-Preserving Generalized Derivations on Semiprime Rings, Acta Mathematica Sinica 24 (11) (2008) 1835–1842.
4. E. Koç, Ö. Gölbaşı, Some Results on Ideals of Semiprime Rings with Multiplicative Generalised Derivations, Communication in Algebra 46 (11) (2018) 4905–4913.
5. A. Ali, M. Yasen, M. Anwar, Strong Commutativity Preserving Mappings on Semiprime Rings, Bulletin Korean Mathematical Society 43 (4) (2006) 711–713.
6. M. S. Samman, On Strong Commutativity-Preserving Maps, International Journal of Mathematics and Mathematical Sciences 6 (2005) 917–923.
7. A. Melaibari, N. Muthana, A. Al-Kenani, Homoderivations on Rings, General Mathematics Notes 35 (1) (2016) 1–8.
8. I. N. Herstein, A Note on Derivations, Canadian Mathematical Bulletin 21 (3) (1978) 369–370.

9. J. Bergen, I. N. Herstein, J. W. Kerr, Lie Ideals and Derivation of Prime Rings, Journal of Algebra 71 (1981) 259–267.
10. E. Koç, Ö. Gölbaşı, On (σ,τ)-Lie Ideals with Generalised Derivation, Bulletin of the Korean Mathematical Society 47 (6) (2010) 1121–1129.
11. M. Ashraf, A. Ali, S. Ali, Some Commutativity Theorems for Rings with Generalised Derivations, Southeast Asian Bulletin Mathematics 31 (2007) 415–421.
12. M. M. El Sofy Aly, Rings with Some Kinds of Mappings, Master’s Thesis Cairo University (2000) Cairo.
13. M. N. Daif, H. E. Bell, Remarks on Derivations on Semiprime Rings, International Journal of Mathematics and Mathematical Sciences 15 (1) (1992) 205–206.
14. E. F. Alharfie, N. M. Muthana, The Commutativity of Prime Rings with Homoderivations, International Journal of Advanced and Applied Sciences 5 (5) (2018) 79–81.
15. N. Rehman, M. R. Mozumder, A. Abbasi, Homoderivations on Ideals of Prime and Semiprime Rings, The Aligarh Bulletin of Mathematics 38 (1-2) (2019) 77–87.
16. M. S. El-Sayiad, A. Ageeb, A. Ghareeb, Centralizing n-Homoderivations of Semiprime Rings, Journal of Mathematics 2022 (2022) Article ID 1112183 8 pages.