# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

## Article

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Bull. Korean Math. Soc. 2022; 59(3): 725-743

Published online May 31, 2022 https://doi.org/10.4134/BKMS.b210427

## A generalization of $w$-linked extensions

Xiaoying Wu

Sichuan Normal University

### Abstract

In this paper, the concepts of $w$-linked homomorphisms, the $w_{\phi}$-operation, and DW${}_{\phi}$ rings are introduced. Also the relationships between $w_{\phi}$-ideals and $w$-ideals over a $w$-linked homomorphism $\phi: R\ra T$ are discussed. More precisely, it is shown that every $w_{\phi}$-ideal of $T$ is a $w$-ideal of $T$. Besides, it is shown that if $T$ is not a DW${}_{\phi}$ ring, then $T$ must have an infinite number of maximal $w_{\phi}$-ideals. Finally we give an application of Cohen's Theorem over $w$-factor rings, namely it is shown that an integral domain $R$ is an SM-domain with $w$-$\dim(R)\leq 1$, if and only if for any nonzero $w$-ideal $I$ of $R$, $(R/I)_w$ is an Artinian ring, if and only if for any nonzero element $a\in R$, $(R/(a))_w$ is an Artinian ring, if and only if for any nonzero element $a\in R$, $R$ satisfies the descending chain condition on $w$-ideals of $R$ containing $a$.

Keywords: $w$-linked homomorphism, $w_{\phi}$-operation, DW${}_{\phi}$ ring, $w$-factor ring

MSC numbers: 13B02, 13E05