Authors:
Yaqoub Ahmed,M. Aslam, Liaqat Ali,DOI NO:
https://doi.org/10.26782/jmcms.2020.07.00011Keywords:
Jordan left derivation,Involution,Prime semirings,Additive Inverse semirings,Abstract
In this article we introduce A*-involution in additively inverse semirings. This involution have potential to extend the striking results of B*-algebras, C*- algebras and involutory rings in the domain of semirings. The remarkable result due to Herstein[XII] states that every Jordan derivation on a 2-torsion free prime ring is a derivation. In the present paper, we shall study the above mentioned result for Jordan left derivations in semirings with A* -Involution.Refference:
I. Awtar, R, Lie ideals and Jordan derivations of prime rings, Proc. Amer. Math. Soc.90 (1984), 9-14.
II. Ashraf, M. and N. Ur. Rehmann, On Lie ideals and Jordan left derivations of prime rings. Arch. Math. (Brno) 36 (2000), 201-206
III. Bandlet H.J. and M. Petrich, Subdirect products of rings and distributive lattices Proc. Edin Math.Soc. 25 (1982) 135-171.
IV. Beidar K.I, WS Martindale On Functional Identities in Prime Rings with Involution,Journal of Algebra Volume 203. Issue 2, 15 May 1998, 491-532.
V. Bergen, J., Herstein, I.N. and Ker, J.W., Lie ideals and derivations of prime rings,J. Algebra 71 (1981), 259-267.
VI. Bresar. M. and Vukman, J., On left derivations and related mappings, Proc. Ameer. Math. Soc. 110 (1990), 7-16.
VII. C. Lanski, Commutation with skew elements in rings with involution, Pacific J. Math. Volume 83, Number 2(1979), 393-399.
VIII. Chadja. I , H. LANGER , Near Semirings and Semirings with Involution, Miskolc Mathematical Notes, Vol.17 (2017) No. 2, 801810
IX. Fadaee. B and H.Ghahramani, Continuous linear maps on reflexive algebras behaving like Jordan left derivations at idempotent- product elements,ar Xiv:1312.6953
X. Ghahramani. H., Characterizations of left derivable maps at non- trivial idempotents on nestalgebras, arXiv:1312.6959.
XI. Golan. J.S., The theory of semirings with applications in mathematics and theoretical computer science (John Wiley and Sons . Inc, New York, 1992). doi:10.1007/978- 94-015-9333-5-13.
XII. Herstein, I.N, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8(1957), 1104- 1110.
XIII. J. Li and J. Zhou, Jordan left derivations and some left derivable maps, Oper. Matrices 4(2010),127-138.
XIV. J.V. Markov, Pierce Sheaf for semirings with involution, Russian Mathmatics (Iz. VUZ), 2014, Vol.58, No. 4, 1419.
XV. Javed. M. A, Aslam M. Hussain M., On condition (A 2) of Bandlet and Petrich for inverse semirings, Int. Math. Forum, 2012,7, 2903-2914.
XVI. Javed. M. A., Aslam M., Some commutativity conditions in prime MA-semirings, Ars Combin., 2014,114,373-384.
XVII. Karvellas P.H, Inversive semirings, J. Austral. Math. Soc., 1974,18,277-288.
XVIII. Kill-Wong Jun and Byung-Do Kim, A note on Jordan left derivations, Bull. Korean Math. Soc.33 (1996) No.2,221-228.
XIX. M. Bresar, Characterizing homomorphisms, multopliers and derivations in rings with idempotents, Proc. Roy,Soc.Edinburgh Sect. A137(2007), 9-21.
XX. M. Burgos, J. Cabello-Sanchezanda. M . Peralta, Linear maps between C*-algebras that are*- homomorohism at a fixed point, arXiv: 1609.07776.
XXI. M.A. Chebotar, W.F. Ke and P.H .Lee , Maps characterized by action on zero products, Pacific J. Math.216 (2004), 217-2278.
XXII. Oukhtite.L., S. Salhi , On commutativity of – prime rings. Glas. Math. Ser. III Vol.41, no.1 (2006), 57-64.
XXIII. Yaqoub Ahmed, W.A. Dudek, Stronger Lie derivations on MA-semirings, Afrika Mat., doi.org/10.1007/s13370-020-00768-3.
XXIV. Yaqoub Ahmed, W.A. Dudek. Left Jordan derivation on certain semirings,. Hacepette J. Math. (accepted).
XXV. Vukman. J. On left Jordan derivations of rings and Banach algebras, Equations Math .75 (2008), no. 3, 260-266
XXVI. Y. Ahmed, M. Nadeem, M. Aslam, On Centralizers of MA semirings, JMCMS, Vol 15 (4), 47-57
XXVII. Y. Ahmed, Wieslaw Dudek, M. Aslam, Asian European journal of Mathematics (accepted), DOI: 10.1142/S1793557121500674