Quotient rings satisfying some identities

Mohammadi El Hamdaoui; Abdelkarim Boua

doi:10.56754/0719-0646.2503.455

Authors

Mohammadi El Hamdaoui Department of Mathematics, Polydisciplinary Faculty, LSI, Taza, Sidi Mohammed Ben Abdellah University, Fes, Morocco.
Abdelkarim Boua Department of Mathematics, Polydisciplinary Faculty, LSI, Taza, Sidi Mohammed Ben Abdellah University, Fes, Morocco. https://orcid.org/0000-0002-6397-4713

DOI:

https://doi.org/10.56754/0719-0646.2503.455

Keywords:

Derivations, prime ideals, prime rings

Abstract

This paper investigates the commutativity of the quotient ring \(\mathcal{R}/P\), where \(\mathcal{R}\) is an associative ring with a prime ideal \(P\), and the possibility of forms of derivations satisfying certain algebraic identities on \(\mathcal{R}\). We provide some results for strong commutativity-preserving derivations of prime rings.

Downloads

Download data is not yet available.

References

A. Ali, M. Yasen, and M. Anwar, “Strong commutativity preserving mappings on semiprime rings,” Bull. Korean Math. Soc., vol. 43, no. 4, pp. 711–713, 2006, doi: 10.4134/BKMS.2006.43.4.711.

F. A. A. Almahdi, A. Mamouni, and M. Tamekkante, “A generalization of Posner’s Theorem on derivations in rings,” Indian J. Pure Appl. Math., vol. 51, no. 1, pp. 187–194, 2020, doi: https://doi.org/10.1007/s13226-020-0394-8.

K. I. Beidar, W. S. Martindale, III, and A. V. Mikhalev, Rings with generalized identities, ser. Monographs and Textbooks in Pure and Applied Mathematics. New York, USA: Marcel Dekker, Inc., 1996, vol. 196.

H. E. Bell and M. N. Daif, “On commutativity and strong commutativity-preserving maps,” Canad. Math. Bull., vol. 37, no. 4, pp. 443–447, 1994, doi: https://doi.org/10.4153/CMB- 1994-064-x.

H. E. Bell and G. Mason, “On derivations in near-rings and rings,” Math. J. Okayama Univ., vol. 34, pp. 135–144, 1992.

M. Brešar, “Semiderivations of prime rings,” Proc. Amer. Math. Soc., vol. 108, no. 4, pp. 859–860, 1990, doi: 10.2307/2047937.

M. Brešar, “On the distance of the composition of two derivations to the generalized derivations,” Glasgow Math. J., vol. 33, no. 1, pp. 89–93, 1991, doi: 10.1017/S0017089500008077.

M. Brešar, “Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings,” Trans. Amer. Math. Soc., vol. 335, no. 2, pp. 525–546, 1993, doi: 10.2307/2154392.

M. Brešar and C. R. Miers, “Strong commutativity preserving maps of semiprime rings,” Canad. Math. Bull., vol. 37, no. 4, pp. 457–460, 1994, doi: 10.4153/CMB-1994-066-4.

Q. Deng and M. Ashraf, “On strong commutativity preserving mappings,” Results Math., vol. 30, no. 3-4, pp. 259–263, 1996, doi: 10.1007/BF03322194.

T.-K. Lee and T.-L. Wong, “Nonadditive strong commutativity preserving maps,” Comm. Algebra, vol. 40, no. 6, pp. 2213–2218, 2012, doi: 10.1080/00927872.2011.578287.

J.-S. Lin and C.-K. Liu, “Strong commutativity preserving maps on Lie ideals,” Linear Algebra Appl., vol. 428, no. 7, pp. 1601–1609, 2008, doi: 10.1016/j.laa.2007.10.006.

J.-S. Lin and C.-K. Liu, “Strong commutativity preserving maps in prime rings with involution,” Linear Algebra Appl., vol. 432, no. 1, pp. 14–23, 2010, doi: 10.1016/j.laa.2009.06.036.

C.-K. Liu, “Strong commutativity preserving generalized derivations on right ideals,” Monatsh. Math., vol. 166, no. 3-4, pp. 453–465, 2012, doi: 10.1007/s00605-010-0281-1.

C.-K. Liu and P.-K. Liau, “Strong commutativity preserving generalized derivations on Lie ideals,” Linear Multilinear Algebra, vol. 59, no. 8, pp. 905–915, 2011, doi: 10.1080/03081087.2010.535819.

J. Ma, X. W. Xu, and F. W. Niu, “Strong commutativity-preserving generalized derivations on semiprime rings,” Acta Math. Sin. (Engl. Ser.), vol. 24, no. 11, pp. 1835–1842, 2008, doi: 10.1007/s10114-008-7445-0.

E. C. Posner, “Derivations in prime rings,” Proc. Amer. Math. Soc., vol. 8, pp. 1093–1100, 1957, doi: 10.2307/2032686.

M. S. Samman, “On strong commutativity-preserving maps,” Int. J. Math. Math. Sci., vol. 2005, no. 6, pp. 917–923, 2005, doi: 10.1155/IJMMS.2005.917.

P. Šemrl, “Commutativity preserving maps”, Linear Algebra Appl., vol. 429, no. 5-6, pp. 1051– 1070, 2008, doi: 10.1016/j.laa.2007.05.006.