Left Jordan derivations on certain semirings
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 $S$.
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- [1] Y. Ahmed and W.A. Dudek, Stronger Lie derivations on $MA$-semirings, Afrika Mat.
31 (5-6), 891-901, 2020.
- [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] M. Ashraf and N.U. Rehmann, On Lie ideals and Jordan left derivations of prime
rings, Arch. Math. (Brno) 36, 201-206, 2000.
- [4] M. Ashraf and N.U. Rehmann, On commutativity of rings with derivations, Results
Math. 42, 3-8, 2002.
- [5] R. Awtar, Lie ideals and Jordan derivations of prime rings, Proc. Amer. Math. Soc.
90, 9-14, 1984.
- [6] H.E. Bell and J. Lucier, On additive maps and commutativity in rings, Results Math.
36, 1-8, 1999.
- [7] M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer.
Math. Soc., 110, 7-16, 1990.
- [8] W.A. Dudek, M. Shabir and R. Anjum, Characterizations of hemirings by their hideals,
Comput. Math. Appl. 59 (9), 3167-3179, 2010.
- [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] J.S. Golan, Semirings and their Applications, Kluwer Acad. Publ. 1999.
- [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] P.H. Karvellas, Inversive semirings, J. Austral. Math. Soc. 18, 277-288, 1974.
- [13] M. Nadeem, M. Aslam, On the generalization of Brešar theorems, Quasigroups and
Related Systems 24, 123-128, 2016.
- [14] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093-1100, 1957.
- [15] S. Shafiq and M. Aslam, Centralizers on semiprime MA-semirings, Quasigroups and
Related Systems 24, 269-276, 2016.
- [16] S. Shafiq and M. Aslam, On Jordan mappings of inverse semirings, Open Math. 17,
1123-1131, 2017.
- [17] J. Vukman, On left Jordan derivations of rings and Banach algebras, Aequationes
Math. 75, 260-266 2008.
- [18] J. Zhan and W.A. Dudek, Fuzzy h-ideals of hemirings, Inform. Sci. 177 (3), 876-886,
2007.