Pr\"ufer $v$-multiplication domains and related domains of the form $D+E[\Gamma^*]$

Abstract

Let $D \subseteq E$ be an extension of integral domains, $K$ bethe quotient field of $D$, $S$ be a(saturated) multiplicative subset of $D$ with $D \subsetneq D_S$,$\Gamma$ be a nonzero torsion-free (additive) grading monoid with$\Gamma \cap -\Gamma =\{0\}$, $\Gamma^*=\Gamma \setminus \{0\}$,$G$ be the quotient group of $\Gamma$, $D[\Gamma]$ be the semigroup ring of $\Gamma$ over $D$,$D^{(S, \Gamma)}=D+D_S[\Gamma^*]=\{f \in D_S[\Gamma] \mid f(0) \in D\}$, and$(D, E, \Gamma)=D+E[\Gamma^*]=\{f \in E[\Gamma] \mid f(0) \in D\}$.In this dissertation,we study P$v$MDs and related domains in the view of the composite semigroup ring$(D, E, \Gamma)$. To do that, we first investigate the domain $D^{(S, \Gamma)}$as a special case of $(D, E, \Gamma)$.In fact, we show that $D^{(S, \Gamma)}$ is a P$v$MD (resp.,GCD-domain, GGCD-domain, integrally closed AGCD-domain) if and only if$D$ is a P$v$MD (resp., GCD-domain, GGCD-domain, integrally closed AGCD-domain), $\Gamma$ isa valuation semigroup and $S$ is a $t$-splitting (resp., splitting,$d$-splitting, almost splitting) set of $D$. We also prove that$D^{(S, \Gamma)}$ is a Pr\"ufer domain (resp., B\'ezout domain) if and only if$D$ is a Pr\"ufer domain (resp., B\'ezout domain), $\Gamma$ is a Pr\"ufer submonoid of$\mathbb{Q}$ and $D_S=K$.We also give some examples of a nonzero torsion-free (additive) grading valuation semigroup.Next, by using these results, we study the domain $(D, E, \Gamma)$.We prove that if $E \cap K=D$, then $(D, E, \Gamma)$ is a P$v$MD (resp., GCD-domain, GGCD-domain,Pr\"ufer domain) if and only if $D=E$, $D$ is a P$v$MD (resp., GCD-domain, GGCD-domain,field) and $\Gamma$ is a P$v$MS (resp., GCD-semigroup, GGCD-semigroup, Pr\"ufer submonoid of $\mathbb{Q}$).We show that if $D \subsetneq K \subseteq E$, then$(D, E, \Gamma)$ is a P$v$MD (resp., GCD-domain, GGCD-domain, integrally closed AGCD-domain)if and only if $D$ is a P$v$MD (resp., GCD-domain, GGCD-domain, integrally closed AGCD-domain),$E=K$ and $\Gamma$ is a valuation semigroup.We also show that if $D \subsetneq K \subseteq E$, then$(D, E, \Gamma)$ is a B\'ezout domain (resp., Pr\"ufer domain) if and only if $D$ isa B\'ezout domain (resp., Pr\"ufer domain), $E=K$ and $\Gamma$ is a Pr\"ufer submonoid of $\mathbb{Q}$.For the general case, we prove that if $D \subsetneq E$, then$(D, E, \Gamma)$ is a GCD-domain (resp., integrally closed AGCD-domain) if and only if$D$ is a GCD-domain (resp., integrally closed AGCD-domain), $\Gamma$ is a valuation semigroup and$E=D_S$ for some splitting (resp., almost splitting) set $S$ of $D$.Also, we show that if $D \subsetneq E$, then $(D, E, \Gamma)$ is a B\'ezout domain if and only if$D$ is a B\'ezout domain, $E=K$ and$\Gamma$ is a Pr\"ufer submonoid of $\mathbb{Q}$. Finally,we characterize generalized Krull domains and GUFDs via the domain $(D, E, \Gamma)$.We prove that if $G$ is of type $(0, 0, 0, \cdots)$, then $(D, E, \Gamma)$ is a generalized Krull domain (resp.,GUFD) if and only if $D=E$, $D$ is a generalized Krull domain (resp., GUFD)and $\Gamma$ is a generalized Krull semigroup (resp., weakly factorial GCD-semigroup).