Abstract
A ring R is called right Ikeda–Nakayama ring (right IN-ring) if for any two right ideals I, J of R, \(l(I)+l(J)=l(I \cap J)\). In this paper, we introduce the concept of Essential Ikeda–Nakayama rings (EIN-rings) as a generalization of right IN-rings. This class of rings includes semiprime rings. We prove that for a left nonsingular EIN-ring R, closed ideals of R are right annihilator in R. We show that the class of EIN-rings is closed under direct product and upper triangular matrix rings. Furthermore, a ring R is an Armendariz EIN-ring if and only if R[x] is an EIN-ring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Camillo, V., Nicholson, W.K., Yousif, M.F.: Ikeda–Nakayama rings. J. Algebra 226, 1001–1010 (2000)
Cannon, G.A., Neuerburg, K.M.: Ideals in Dorroh extensions of rings. Missouri J. Math. Sci. 20(3), 165–168 (2008)
Lam, T.Y.: Lectures on Modules and Rings. Springer, New York (1999)
Rege, M.B., Chhawchharia, S.: Armendariz rings. Proc. Jpn. Acad. Ser. A Math. Sci. 73, 14–17 (1997)
Acknowledgements
This paper is supported by Islamic Azad University Central Tehran Branch (IAUCTB). The authors want to thank the authority of IAUCTB for their support to complete this research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Derakhshan, M., Sahebi, S. & Haj Seyed Javadi, H. A note on essential Ikeda–Nakayama rings. Rend. Circ. Mat. Palermo, II. Ser 71, 145–151 (2022). https://doi.org/10.1007/s12215-021-00610-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-021-00610-0