Euclidean Modules

Jichun Liu^ab, Miaosen Chen<sup>c

a</sup> Zhejiang University
^b Fujian Jiangxia University
^c Zhejiang Normal University

Abstract: In this paper, we introduce the notion of Euclidean module and weakly Euclidean ring. We prove the main result that a commutative ring $R$ is weakly Euclidean if and only if every cyclic $R$-module is Euclidean, and also if and only if $\operatorname{End}({}_{R}M)$ is weakly Euclidean for each cyclic $R$-module $M$. In addition, some properties of torsion-free Euclidean modules are presented.

Keywords: Euclidean modules, weakly Euclidean rings, torsion-free Euclidean modules.

UDC: 512.533

Received: 04.06.2012
Revised: 15.03.2013

DOI: 10.4213/mzm10455

 English version: Mathematical Notes, 2014, 95:6, 865–872

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