Ali N.A. Koam, Moin A. Ansari, Shakir Ali

On $^*$-differential identities in prime rings with involution

Let $\mathcal{R}$ be a ring. An additive map $x\mapsto x^*$ of $\mathcal{R}$ into itself is called an involution if (i) $(xy)^*=y^*x^*$ and (ii) $(x^*)^*=x$ hold for all $x,y\in \mathcal{R}$. In this paper, we study the effect of involution $"*"$ on prime rings that satisfying certain differential identities. The identities considered in this manuscript are new and interesting. As the applications, many known theorems can be either generalized or deduced. In particular, a classical theorem due to Herstein [A note on derivation II, Canad. Math. Bull., 1979] is deduced.

Keywords:

prime ring, commutativity involution, derivation, *-differential identities,

PDF

___

[1] S. Ali and H. Alhazmi, Some commutativity theorems in prime rings with involution and derivations, J. Adv. Math. Comput. Sci. 24 (5), 1–6, 2017.
[2] S. Ali and N.A. Dar, On $*$-centralizing mappings in rings with involution, Georgian Math. J. 1, 25–28, 2014.
[3] S. Ali and S. Huang, On derivations in semiprime rings, Algebr. Represent. Theory 15 (6), 1023–1033, 2012.
[4] S. Ali, N.A. Dar, and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J. 23 (1), 9–14, 2016.
[5] S. Ali, M.S. Khan, and M. Al-Shomrani, Generalization of Herstein theorem and its applications to range inclusion problems, J. Egyptian Math. Soc. 22, 322–326, 2014.
[6] N. Argac, On prime and semiprime rings with derivations, Algebra Colloq. 13 (3), 371–380, 2006.
[7] M. Ashraf and M.A. Siddeeque, On $*-$n-derivations in prime rings with involution, Georgian Math. J. 21 (1), 9–18, 2014.
[8] M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math. 42 (1-2), 3–8, 2002.
[9] H.E. Bell, On the commutativity of prime rings with derivation, Quaest. Math. 22, 329-333, 1991.
[10] H.E. Bell and M.N. Daif, On derivations and commutativity in prime rings, Acta Math. Hungar. 66, 337–343, 1995.
[11] M.N. Daif, Commutativity results for semiprime rings with derivation, Int. J. Math. Math. Sci. 21 (3), 471–474, 1998.
[12] N.A. Dar and S. Ali, On $*$-commuting mappings and derivations in rings with involution, Turk. J. Math. 40, 884–894, 2016.
[13] V.De. Filippis, On derivation and commutativity in prime rings, Int. J. Math. Math. Sci. 69-72, 3859–3865, 2004.
[14] A. Fosner and J. Vukman, Some results concerning additive mappings and derivations on semiprime rings, Pul. Math. Debrecen, 78 (3-4), 575–581, 2011.
[15] I.N. Herstein, Rings with Involution, University of Chicago Press, Chicago, 1976.
[16] I.N. Herstein, A note on derivation II, Canad. Math. Bull. 22, 509–511, 1979.
[17] J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19, 113– 117, 1976.
[18] L. Oukhtite, Posner’s second theorem for Jordan ideals in ring with involution, Expo. Math. 4 (29), 415–419, 2011.
[19] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093–1100, 1957.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

Arşiv