J. Korean Math. Soc. 2015; 52(1): 1-21
Printed January 1, 2015
https://doi.org/10.4134/JKMS.2015.52.1.1
Copyright © The Korean Mathematical Society.
Alfred Geroldinger, Sebastian Ramacher, and Andreas Reinhart
Karl--Fran\-zens--Universit\"at Graz, NAWI Graz, Karl--Fran\-zens--Universit\"at Graz, NAWI Graz, Karl--Fran\-zens--Universit\"at Graz, NAWI Graz
C-domains are defined via class semigroups, and every C-domain is a Mori domain with nonzero conductor whose complete integral closure is a Krull domain with finite class group. In order to extend the concept of C-domains to rings with zero divisors, we study $v$-Marot rings as generalizations of ordinary Marot rings and investigate their theory of regular divisorial ideals. Based on this we establish a generalization of a result well-known for integral domains. Let $R$ be a $v$-Marot Mori ring, $\widehat R$ its complete integral closure, and suppose that the conductor $\mathfrak f = (R : \widehat R)$ is regular. If the residue class ring $R/\mathfrak f$ and the class group $\mathcal C (\widehat R)$ are both finite, then $R$ is a C-ring. Moreover, we study both $v$-Marot rings and C-rings under various ring extensions.
Keywords: Marot rings, Mori rings, Krull rings, Krull monoids, C-rings, C-monoids
MSC numbers: 13F05, 13A15, 20M14
2019; 56(4): 869-915
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