Prime rings with involution involving left multipliers
Keywords:Prime ring, Derivation, Multiplier, Involution, Commutativity
AbstractLet R be a prime ring of characteristic different from 2 with involution ’?’ of the second kind and n ? 1 be a fixed positive integer. In the present paper it is shown that if R admits nonzero left multipliers S and T, then the following conditions are equivalent: (i) R is commutative. (ii) Tn([x, x?]) 2 Z(R) for all x 2 R; (iii) Tn(x ? x?) 2 Z(R) for all x 2 R; (iv) [S(x), T(x?)] 2 Z(R) for all x 2 R; (v) [S(x), T(x?)] - (x ? x?) 2 Z(R) for all x 2 R; (vi) S(x) ? T(x?) 2 Z(R) for all x 2 R; (vii) S(x) ? T(x?) - [x, x?] 2 Z(R) for all x 2 R. The existence of hypotheses in various theorems have been justified by the examples.
S. Ali and N. A. Dar, “On centralizers of prime rings with involution”, Bulletin of the Iranian Mathematical Society, vol. 41, no. 6, pp. 1465-1475, 2015. [On line]. Available: https://bit.ly/2VZy42o
M. Ashraf and N.-U. Rehman, “On commutativity of rings with derivations”, Results in mathematics, vol. 42, no. 1-2, pp. 3–8, Aug. 2002, doi: 10.1007/BF03323547.
M. Ashraf and S. Ali, “On left multipliers and the commutativity of prime rings”, Demonstratio mathematica, vol. 41, no. 4, Oct. 2008, doi: 10.1515/dema-2008-0404.
N. Divinsky, “On commuting automorphisms of rings”, Transactions of the Royal Society of Canada Section III, vol. 49, no. 3, pp. 19-22, 1955.
I. R. Hentzel and M. T. El-Sayiad, “Left centralizers on rings that are not semiprime”, Rocky mountain journal of mathematics, vol. 41, no. 5, pp. 1471–1482, 2011, doi: 10.1216/rmj-2011-41-5-1471.
J. Luh, “A Note on Commuting Automorphisms of Rings”, The american mathematical monthly, vol. 77, no. 1, pp. 61–62, Jan. 1970, doi: 10.1080/00029890.1970.11992420.
E. C. Posner, “Derivations in prime rings”, Proceedings of the American Mathematical Society, vol. 8, no. 6, pp. 1093–1100, Jun. 1957, doi: 10.1090/S0002-9939-1957-0095863-0.
J. Vukman and I. Kosi-Ulbl, “On centralizers of semiprime rings”, Aequationes mathematicae, vol. 66, no. 3, pp. 277–283, Dec. 2003, doi: 10.1007/s00010-003-2681-y.
B. Zalar, “On centralizer of semiprime rings”, Commentationes mathematicae Universitatis Carolinae, vol. 32, no. 4, pp. 609-614, 1991. [On line]. Available: https://bit.ly/2KxlhyD