Left Jordan derivations on certain semirings

Abstract

We determine conditions under which a left Jordan derivation defined on an $MA$-semiring $S$ is a left derivation on this semiring and prove when a left Jordan derivation on $S$ implies the commutativity of $S$.

Keywords

• Left derivation
• Jordan derivation
• inverse semiring
• torsion free semiring

References

1. [1] Y. Ahmed and W.A. Dudek, Stronger Lie derivations on $MA$-semirings, Afrika Mat. 31 (5-6), 891-901, 2020.
2. [2] S. Ali, M. Ashraf, M.S. Khan and J. Vukman, Commutativity of rings involving additive mappings, Quaest. Math. 37 (2), 215-229, 2014.
3. [3] M. Ashraf and N.U. Rehmann, On Lie ideals and Jordan left derivations of prime rings, Arch. Math. (Brno) 36, 201-206, 2000.
4. [4] M. Ashraf and N.U. Rehmann, On commutativity of rings with derivations, Results Math. 42, 3-8, 2002.
5. [5] R. Awtar, Lie ideals and Jordan derivations of prime rings, Proc. Amer. Math. Soc. 90, 9-14, 1984.
6. [6] H.E. Bell and J. Lucier, On additive maps and commutativity in rings, Results Math. 36, 1-8, 1999.
7. [7] M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc., 110, 7-16, 1990.
8. [8] W.A. Dudek, M. Shabir and R. Anjum, Characterizations of hemirings by their hideals, Comput. Math. Appl. 59 (9), 3167-3179, 2010.

9. [9] K. Glazek, A Guide to Literature on Semirings and their Applications in Mathematics and Information Sciences with Complete Bibliography, Kluwer Acad. Publ., Dodrecht, 2002.
10. [10] J.S. Golan, Semirings and their Applications, Kluwer Acad. Publ. 1999.
11. [11] M.A. Javed, M. Aslam and M. Hussain, On condition $(A_{2})$ of Bandelt and Petrich for inverse semirings, Int. Math. Forum 7, 2903-2914, 2012.
12. [12] P.H. Karvellas, Inversive semirings, J. Austral. Math. Soc. 18, 277-288, 1974.
13. [13] M. Nadeem, M. Aslam, On the generalization of Brešar theorems, Quasigroups and Related Systems 24, 123-128, 2016.
14. [14] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093-1100, 1957.
15. [15] S. Shafiq and M. Aslam, Centralizers on semiprime MA-semirings, Quasigroups and Related Systems 24, 269-276, 2016.
16. [16] S. Shafiq and M. Aslam, On Jordan mappings of inverse semirings, Open Math. 17, 1123-1131, 2017.
17. [17] J. Vukman, On left Jordan derivations of rings and Banach algebras, Aequationes Math. 75, 260-266 2008.
18. [18] J. Zhan and W.A. Dudek, Fuzzy h-ideals of hemirings, Inform. Sci. 177 (3), 876-886, 2007.