ENDOMORPHISMS WITH CENTRAL VALUES ON PRIME RINGS WITH INVOLUTION

Year 2020, Volume: 28 Issue: 28, 127 - 140, 14.07.2020

L. Oukhtıte H. El Mır B. Nejjar

https://doi.org/10.24330/ieja.768202

Cited By: 2

Abstract

In this paper we present some commutativity theorems for prime rings $R$ with involution $\ast$ of the second kind in which endomorphisms satisfy certain algebraic identities. Furthermore, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

Keywords

Prime ring, involution, commutativity, endomorphism

References

• S. Ali and S. Huang, On derivations in semiprime rings, Algebr. Represent. Theory, 15(6) (2012), 1023-1033.
• M. Ashraf and M. A. Quadri, Some conditions for the commutativity of rings, Acta Math. Hungar., 61(1-2) (1993), 73-77.
• M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math., 42(1-2) (2002), 3-8.
• H. E. Bell, Some results on commutativity and anti-commutativity in rings, Acta Math. Hungar., 63(2) (1994), 113-117.
• H. E. Bell, A. Boua and L. Oukhtite, Semigroup ideals and commutativity in 3-prime near rings, Comm. Algebra, 43(5) (2015), 1757-1770.
• H. E. Bell and M. N. Daif, On commutativity and strong commutativity-preserving maps, Canad. Math. Bull., 37(4) (1994), 443-447.
• H. E. Bell and M. N. Daif, On derivations and commutativity in prime rings, Acta Math. Hungar., 66(4) (1995), 337-343.
• H. E. Bell and W. S. Martindale, III, Centralizing mappings of semiprime rings, Canad. Math. Bull., 30(1) (1987), 92-101.
• M. Bresar, Commuting traces of biadditive mapping, commutativity preserving mapping and Lie mappings, Trans. Amer. Math. Soc., 335(2) (1993), 525-546.
• M. Bresar, On the distance of the composition of two derivations to the generalized derivation, Glasgow Math. J., 33(1) (1991), 89-93.
• M. Bresar and C. R. Miers, Strong commutativity preserving maps of semiprime rings, Canad. Math. Bull., 37(4) (1994), 457-460.
• V. De Filippis, A. Mamouni and L. Oukhtite, Generalized Jordan semiderivations in prime rings, Canad. Math. Bull., 58(2) (2015), 263-270.
• Q. Deng and M. Ashraf, On strong commutativity preserving mappings, Results Math., 30 (1996), 259-263.
• N. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III, 49 (1955), 19-22.
• A. Fosner, X.-F. Liang and F. Wei, Centralizing traces with automorphisms on triangular algebras, Acta Math. Hungar., 154(2) (2018), 315-342.
• A. Mamouni, B. Nejjar and L. Oukhtite, Differential identities on prime rings with involution, J. Algebra Appl., 17(9) (2018), 1850163 (11 pp).
• A. Mamouni, L. Oukhtite and B. Nejjar, On $\ast$-semiderivations and $\ast$-generalized semiderivations, J. Algebra Appl., 16(4) (2017), 1750075 (8 pp).
• B. Nejjar, A. Kacha, A. Mamouni and L. Oukhtite, Commutativity theorems in rings with involution, Comm. Algebra, 45(2) (2017), 698-708.
• L. Oukhtite, Posner's second theorem for Jordan ideals in rings with involution, Expo. Math., 29(4) (2011), 415-419.
• L. Oukhtite and A. Mamouni, Generalized derivations centralizing on Jordan ideals of rings with involution, Turkish J. Math., 38(2) (2014), 225-232.
• E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.
• P. Semrl, Commutativity preserving maps, Linear Algebra Appl., 429 (2008), 1051-1070.