On ideals of prime rings involving $n$-skew commuting additive mappings with applications

Abstract

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.

Keywords

• ideal
• prime ring
• C*-algebra
• commuting mapping
• n-skew-commuting mapping

References

1. [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. [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. [3] S. Ali and N.A. Dar, On ∗-centralizing mappings in rings with involution, Georgian Math. J. 21(1), 25-28, 2014.
4. [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. [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. [6] P. Ara and M. Mathieu, Local multipliers of $C^*$-algebras, Springer Monograph in Mathematics, Springer-Verlag, London, 2003.
7. [7] M. Ashraf and J. Vukman, On derivations and commutativity in semiprime rings, Aligarh Bull. Math. 18, 29-38, 1999.
8. [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. [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. [10] H.E. Bell and J. Lucier, On additive mappings and commutativity in rings, Result Math. 36, 1-8, 1999.
11. [11] H.E. Bell and W.S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1), 92-101, 1987.
12. [12] M. Brešar, Centralizing mappings on von Neumann algebras, Proc. Amer. Math. Soc., 111 (2), 501-510, 1991.
13. [13] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156, 385-394, 1993.
14. [14] M. Brešar, On skew-commuting mapping of rings, Bull. Austral. Math. Soc. 41, 291- 296, 1993.
15. [15] M. Brešar, Applying the theorem on functional identies, Nova. J. Math. Game. Theory Algebra, 4, 43-54, 1995.
16. [16] M. Brešar, Commuting maps: A Survey, Taiwanese J. Math. 8 (3), 361-397, 2004.
17. [17] M. Brešar and B. Hvala, On additive maps of prime rings, Bull. Austral. Math. Soc. 51, 377-381, 1995.
18. [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. [19] L.O. Chung and J. Luh, On semicommuting automorphisms of rings, Canad. Math. Bull. 21, 13-16, 1978.
20. [20] L.O. Chung and J. Luh, Semiprime rings with nilpotent derivations, Canad. Math. Bull. 24, 415-421, 1981.
21. [21] Q. Deng, On N-centralizing mappings of prime rings, Proc. Roy. Irish Acad. Sect. A, 93 (2), 171-176, 1993.
22. [22] Q. Deng and H.E. Bell, On derivations and commutativity in semiprime rings, Comm. Algebra, 23 (10), 3705-3713, 1995.
23. [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. [24] M. Fošner, Result concerning additive mappings in semiprime rings, Math. Slov. 65, 1271-1276, 2015.
25. [25] A. Fošner and N. Rehman, Identities with additive mappings in semiprime rings, Bull. Korean Math. Soc. 51, 207-211, 2014.
26. [26] A. Fošner and J. Vukman, Some results concerning additive mappings and derivations on semiprime rings, Publ. Math. Debrecen, 78, 575-581, 2011.
27. [27] Y. Hirano, A. Kaya and H. Tominaga, On a theorem of Mayne, Math. J. Okayama Univ. 25(2), 125-132, 1983.
28. [28] A. Kaya, A theorem on semi-centralizing derivations of prime rings, Math. J. Okayama Univ. 27, 11-12, 1985.
29. [29] A. Kaya and C. Koc, Semi-centralizing automorphisms of prime rings, Acta Math. Acad. Sci. Hungar. 38, 53-55, 1981.
30. [30] T.K. Lee and T.C. Lee, Commuting additive mappings in semiprime rings, Bull. Inst. Math. Acad. Sinica 24, 259-268, 1996.
31. [31] J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19, 113- 115, 1976.
32. [32] J. Mayne, Centralizing automorphisms of Lie ideals in prime rings, Canad. Math. Bull. 35, 510-514, 1992.
33. [33] G.J. Murphy, $C^*$-algebras and operator theory, Academic Press INC., New York, 1990.
34. [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. [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. [36] E.C. Posner, Derivation in prime rings, Proc. Amer. Math. Soc. 8, 1093-1100, 1957.
37. [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. [38] R.K. Sharma and B. Dhara, Skew commuting and commuting mappings in rings with left identity, Result. Math. 46, 123-129, 2004.
39. [39] J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109, 47-52, 1990.