Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2022; 37(1): 45-56

Online first article July 8, 2021      Printed January 31, 2022

https://doi.org/10.4134/CKMS.c200467

Copyright © The Korean Mathematical Society.

Rings in which every ideal contained in the set of zero-divisors is a d-ideal

Adam Anebri, Najib Mahdou, Abdeslam Mimouni

University S.M. Ben Abdellah Fez; University S.M. Ben Abdellah Fez; King Fahd University of Petroleum & Minerals

Abstract

In this paper, we introduce and study the class of rings in which every ideal consisting entirely of zero divisors is a d-ideal, considered as a generalization of strongly duo rings. Some results including the characterization of AA-rings are given in the first section. Further, we examine the stability of these rings in localization and study the possible transfer to direct product and trivial ring extension. In addition, we define the class of $d_E$-ideals which allows us to characterize von Neumann regular rings.

Keywords: AA-ring, strongly duo ring, trivial ring extension, localization, direct product, d-ideal, d$_E$-ideal

MSC numbers: Primary 13B99; Secondary 13A15, 13G05, 13B21