Some commutativity criteria involving endomorphism conditions on prime ideals

Year 2022, , 1280 - 1287, 01.10.2022

L. Oukhtite , Abdellah Mamounı , Mohammed Zerra

https://doi.org/10.15672/hujms.918743

Cited By: 1

Abstract

In this paper we initiate a new approach consisting to characterize the commutativity of a quotient ring $R/P$ by endomorphisms of $R$ satisfying some algebraic identities involving the prime ideal $P.$ Some well-known results concerning the commutativity of prime (semi-prime) rings have been improved.

Keywords

prime ideals, commutativity, endomorphisms

References

• [1] C.M. Anwar and A.B. Thaheem, Centralizing mappings and derivations on semiprime rings, Demonstratio Math. 37(2), 285-292, 2004.
• [2] H.E. Bell and M.N. Daif, On commutativity and strong commutativity-preserving maps, Canad. Math. Bull. 37(4), 443-447, 1994.
• [3] I.N. Herstein, A note on derivations, Canad. Math. Bull. 21, 369-370, 1978.
• [4] S. Huang, O. Golbas, and E. Koc, On centralizing and strong commutativity preserving maps of semiprime rings, Ukrainian Math. J. 67(2), 323-331, 2015.
• [5] C. Lanski, Differential identities, Lie ideals and Posner’s theorems, Pacific J. Math. 134(2), 275-297, 1988.
• [6] A. Mamouni, L. Oukhtite and B. Nejjar, Differential identities on prime rings with involution, J. Algebra Appl. 17(9), 1850163, 2018.
• [7] A. Mamouni, L. Oukhtite and B. Nejjar, On ∗-semiderivations and ∗-generalized semiderivations, J. Algebra Appl. 16(4), 1750075, 2017.
• [8] A. Mamouni, L. Oukhtite and M. Zerra, On derivations involving prime ideals and commutativity in rings, São Paulo J. Math. Sci. 14(2), 675-688, 2020.
• [9] A. Mamouni, L. Oukhtite and M. Zerra, Prime ideals and generalized derivations with central values on rings, Rend. Circ. Mat. Palermo (2), 70(3), 1645, 2021.
• [10] J. Mayne, Centralizing automorphisms of prime rings, Can. Math. Bull. 19(1), 113- 115, 1976.
• [11] B. Nejjar, A. Kacha, A. Mamouni and L. Oukhtite, Commutativity theorems in rings with involution, Comm. Algebra 45(2), 698-702, 2017.
• [12] L. Oukhtite and A. Mamouni, Generalized derivations centralizing on Jordan ideals of rings with involution, Turkish J. Math. 38(2), 225-232, 2014.
• [13] L. Oukhtite, A. Mamouni and M. Ashraf, Commutativity theorems for rings with differential identities on Jordan ideals, Comment. Math. Univ. Carolin. 54 (4), 447- 457, 2013.
• [14] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093-1100, 1957.