On ideals of prime rings involving $n$-skew commuting additive mappings with applications
Let $n > 1 $ be a fixed positive integer and $S$ be a subset of a ring $R$. A mapping $\zeta$ of a ring $R$ into itself is called $n$-skew-commuting on $S$ if $\zeta(x)x^{n} + x^{n}\zeta(x)=0$, $\forall$ $x\in S.$ The main aim of this paper is to describe $n$-skew-commuting mappings on appropriate subsets of $R$. With this, many known results can be either generalized or deduced. In particular, this solves the conjecture in [M. Nadeem, M. Aslam and M.A. Javed, On $2$-skew commuting additive mappings of prime rings, Gen. Math. Notes, 2015]. The second main result of this paper is concerned with a pair of linear mappings of $C^*$-algebras. We show that here, if $C^*$-Algebra admits a pair of linear mappings $f$ and $g$ such that $f(x)x^* + x^*g(x) \in Z(A)$ for all $x \in A,$ then both $f$ and $g$ must be zero. As the applications of first main result (Theorem $2.1$) and apart from proving some other results, we characterize the linear mappings on primitive $C^*$-algebras. Furthermore, we provide an example to show that the assumed restrictions cannot be relaxed.
___
- [1] S. Ali, H. Alhazmi, N.A. Dar and A.N. Khan, A characterization of additive mappings
in rings with involution, Algebra & Application, De Gruter Proc. Math. 11-24, 2018.
- [2] S. Ali, M. Ashraf, M.A. Raza and A.N. Khan, N-commuting mappings on (semi)-
prime rings with applications, Comm. Algebra, 47(5), 2262-2270, 2019.
- [3] S. Ali and N.A. Dar, On ∗-centralizing mappings in rings with involution, Georgian
Math. J. 21(1), 25-28, 2014.
- [4] S. Ali, N.A. Dar and J. Vukman, Jordan left ∗-centralizers of prime and semiprime
rings with involution, Beitr. Algebra Geom. 54(2), 609-624, 2013.
- [5] P. Ara and M. Mathieu, An application of local multipliers to centralizing mappings
of $C^*$-algebras, Quart. J. Math. Oxford, 2(44), 129-138, 1993.
- [6] P. Ara and M. Mathieu, Local multipliers of $C^*$-algebras, Springer Monograph in
Mathematics, Springer-Verlag, London, 2003.
- [7] M. Ashraf and J. Vukman, On derivations and commutativity in semiprime rings,
Aligarh Bull. Math. 18, 29-38, 1999.
- [8] K.I. Beidar, Y. Fong, P.-H. Lee and T.-L. Wong, On additive maps of prime rings
satisfying the Engel condition, Comm. Algebra, 25, 3889-3902, 1997.
- [9] K.I. Beidar, W.S. Martindale III and A.V. Mikhalev, Rings with Generalized Identities,
Marcel Dekker, Inc., New York-Basel-Hong Kong, 1996.
- [10] H.E. Bell and J. Lucier, On additive mappings and commutativity in rings, Result
Math. 36, 1-8, 1999.
- [11] H.E. Bell and W.S. Martindale III, Centralizing mappings of semiprime rings, Canad.
Math. Bull. 30 (1), 92-101, 1987.
- [12] M. Brešar, Centralizing mappings on von Neumann algebras, Proc. Amer. Math. Soc.,
111 (2), 501-510, 1991.
- [13] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156,
385-394, 1993.
- [14] M. Brešar, On skew-commuting mapping of rings, Bull. Austral. Math. Soc. 41, 291-
296, 1993.
- [15] M. Brešar, Applying the theorem on functional identies, Nova. J. Math. Game. Theory
Algebra, 4, 43-54, 1995.
- [16] M. Brešar, Commuting maps: A Survey, Taiwanese J. Math. 8 (3), 361-397, 2004.
- [17] M. Brešar and B. Hvala, On additive maps of prime rings, Bull. Austral. Math. Soc.
51, 377-381, 1995.
- [18] M.A. Chaudhary and A.B. Thaheem, A note on a pair of derivations of semiprime
rings, Int. J. Math. Math. Sci. 39, 2097-2102, 2004.
- [19] L.O. Chung and J. Luh, On semicommuting automorphisms of rings, Canad. Math.
Bull. 21, 13-16, 1978.
- [20] L.O. Chung and J. Luh, Semiprime rings with nilpotent derivations, Canad. Math.
Bull. 24, 415-421, 1981.
- [21] Q. Deng, On N-centralizing mappings of prime rings, Proc. Roy. Irish Acad. Sect. A,
93 (2), 171-176, 1993.
- [22] Q. Deng and H.E. Bell, On derivations and commutativity in semiprime rings, Comm.
Algebra, 23 (10), 3705-3713, 1995.
- [23] B. Dhara and S. Ali, On n-centralizing generalized derivations in semiprime rings
with applications to C∗-algebras, J. Algebra Appl. 11 (6), 1250111, 2012.
- [24] M. Fošner, Result concerning additive mappings in semiprime rings, Math. Slov. 65,
1271-1276, 2015.
- [25] A. Fošner and N. Rehman, Identities with additive mappings in semiprime rings, Bull.
Korean Math. Soc. 51, 207-211, 2014.
- [26] A. Fošner and J. Vukman, Some results concerning additive mappings and derivations
on semiprime rings, Publ. Math. Debrecen, 78, 575-581, 2011.
- [27] Y. Hirano, A. Kaya and H. Tominaga, On a theorem of Mayne, Math. J. Okayama
Univ. 25(2), 125-132, 1983.
- [28] A. Kaya, A theorem on semi-centralizing derivations of prime rings, Math. J.
Okayama Univ. 27, 11-12, 1985.
- [29] A. Kaya and C. Koc, Semi-centralizing automorphisms of prime rings, Acta Math.
Acad. Sci. Hungar. 38, 53-55, 1981.
- [30] T.K. Lee and T.C. Lee, Commuting additive mappings in semiprime rings, Bull. Inst.
Math. Acad. Sinica 24, 259-268, 1996.
- [31] J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19, 113-
115, 1976.
- [32] J. Mayne, Centralizing automorphisms of Lie ideals in prime rings, Canad. Math.
Bull. 35, 510-514, 1992.
- [33] G.J. Murphy, $C^*$-algebras and operator theory, Academic Press INC., New York, 1990.
- [34] M. Nadeem, M. Aslam and M.A. Javed, On 2-skew commuting additive mappings of
prime rings, Gen. Math. Notes 31, 1-9, 2015.
- [35] A. Najati and M.M. Saem, Skew-commuting mappings on semiprime and prime rings,
Hacet. J. Math. Stat. 44(4), 887-892, 2015.
- [36] E.C. Posner, Derivation in prime rings, Proc. Amer. Math. Soc. 8, 1093-1100, 1957.
- [37] N. Rehman and V. De Filippis, On n-commuting and n-skew commuting maps with
generalized derivations in prime and semiprime rings, Sib. Math. J. 52(3), 516-523,
2011.
- [38] R.K. Sharma and B. Dhara, Skew commuting and commuting mappings in rings with
left identity, Result. Math. 46, 123-129, 2004.
- [39] J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math.
Soc. 109, 47-52, 1990.