Authors:
Liaqat Ali,Muhammad Aslam,Yaqoub Ahmed Khan,Ghulam Farid,DOI NO:
https://doi.org/10.26782/jmcms.2020.04.00011Keywords:
Inverse semirings,MA-semirings,Generalized derivations,*Jordanideals,Abstract
In this paper we investigate some fundamental results on Jordan ideals, ∗-Jordan ideals, derivations and generalized derivations and hence establish some commutativity results for a certain class of semirings with involutionRefference:
I. B.E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math.,vol. 91,pp: 1-10, 1969.
II. C. Lanski, Commutation with skew elements in rings with involution, Pacific J. Math., vol. 83, no. 2,pp: 393-399, 1979.
III. C.E. Rickart,Banach algebras with an adjoint operation, Ann. Math. (2), vol. 47, no. 3, pp:528-550, 1946.
IV. D.A. Jordan,On the ideals of a Lie algebra of derivations, J. London Math. Soc., vol. 33,no. 2,pp:33-39, 1986.
V. E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., vol. 8, pp:1093-1100, 1957.
VI. H.E. Bell, W. S. Martindale, Centralizing mappings of semiprime rings, Canad. Math. Bull., vol. 30, pp:92-101, 1987.
VII. H.J. Bandlet, M. Petrich, Subdirect products of rings and distributive attics, Proc. Edinburgh Math. Soc., vol. 25, pp:135-171, 1982.
VIII. I.E. Segal, Irreducible representations of operator algebras, Bull. Amer. Math. Soc., vol. 53, pp:73-88, 1947.
IX. I.N. Herstein, Topics in ring theory, Chicago lectures in mathematics, 1969.
X. I.N. Herstein,Rings with involution, University of Chicago, 1976.
XI. J. Berger, I.N. Herstein and J. W. Kerr, Lie ideals ana derivations of prime rings, J. Algebra, vol. 71, pp: 259-267, 1981.
XII. K.I. Beidar, W.S Martindale On functional identities in prime rings with involution, J. Algebra, vol. 203, no.2,pp:491-532, 1998.
XIII. L. Oukhtite, On Jordan ideals and derivations in rings with involution, Comment. Math. Univ. Carolin., 51(3) (2010), 389-395.
XIV. L. Oukhtite, A. Mamouni, Generalized derivations centralizing on Jordan ideals of rings with involution, Turk. J. Math., 38 (2014), 225-232.
XV. L. Oukhtite, A. Mamouni, Derivations satisfying certain algebraic identities on Jordan ideals, Arab. J. Math., vol. 1, no. 3, pp:341-346, 2012.
XVI. L. Oukhtite, Posner’s second theorem for Jordan ideals in rings with involution, Expos. Math., vol. 29, no. 4, pp:415-419, 2011.
XVII. Liaqat Ali, M. Aslam and Yaqoub Ahmed Khan, Commutativity of semirings with involution, Asian-European Journal of Mathematics, (2019) https://doi.org/10.1142/S1793557120501533
XVIII. Liaqat Ali, M. Aslam and Yaqoub Ahmed Khan, On Jordan ideals of inverse semirings with involution, Indian Journal of Science and Technology, vol. 13, no. 4, pp:430–438, 2020.
XIX. Liaqat Ali, M. Aslam and Yaqoub Ahmed Khan, On Posner’s second theorem for semirings with involution, Submitted.
XX. M. Bresar, On the distance of the composition of two derivations to the generalized derivations, Glasg. Math. J., vol. 33, no. 1, pp:89-93, 1991.
XXI. M.A. Javed, M. Aslam and M. Hussain, On condition (A2) of Bandlet and Petrich for inverse semirings, Int. Math. Forum, vol. 7, no. 59, pp:2903-2914, 2012.
XXII. M.N. Daif, Commutativity result for semiprime rings with derivations, Int. J. Math. Math. Sci., vol. 21, no. 3, pp:471-474, 1998.
XXIII. R. Awtar, Lie and Jordan structure in prime rings with derivations, Proc. Am.Math. Soc., vol. 41 ,pp:67-74, 1973.
XXIV. S.M.A. Zaidi, M. Ashraf, S. Ali, On Jordan ideals and left (θ,θ)-derivations in prime rings, Int. J. Math. Math. Sci. vol. 2004, no. 37, pp:1957-1964, 2004.
XXV. Sara Shafiq, M. Aslam, M.A Javed, On centralizer of semiprime inverse semiring, DiscussionesMathematicae, General Algebra andApplications, vol. 36, pp:71-84, 2016.
XXVI. T.K Lee, On derivations of prime rings with involution, Chin. J. Math., vol. 20, no. 2, pp:191-203, 1992.
XXVII. Yaqoub Ahmed Khan, M. Aslam and Liaqat Ali, Commutativity of additive inverse semirings through f(xy) = [x,f(y)], Thai J. of Math., Special Issue,pp:288-300, 2018.