A note on essential Ikeda–Nakayama rings

Mahya Derakhshan | Shervin Sahebi | Hamid Haj Seyyed Javadi

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URL :   http://research.shahed.ac.ir/WSR/WebPages/Report/PaperView.aspx?PaperID=159086
Date :  2021/04/27
Publish in :    Rendiconti del Circolo Matematico di Palermo Series 2
DOI :  https://doi.org/doi.org/10.1007/s12215-021-00610-0
Link :  http://dx.doi.org/doi.org/10.1007/s12215-021-00610-0
Keywords :essential, annihilator

Abstract :

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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∩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 Rx is an EIN-ring.