On v-Marot Mori rings and C-rings

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 introduce $v$-Marot rings as generalizations of ordinary Marot rings and study 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, $\hat R$ its complete integral closure, and suppose that the conductor $\mathfrak f = (R : \hat R)$ is regular. If the residue class ring $R/\mathfrak f$ and the class group $\mathcal C (\hat R)$ are both finite, then $R$ is a C-ring. Moreover, we study both $v$-Marot rings and C-rings under various ring extensions.