Graded integral domains which are UMT-domains

Let $\Gamma$ be a torsionless commutative cancellative monoid, $R =\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain, and $H$ be the set of nonzero homogeneous elements of $R$. In this paper, we show that if $Q$ is a maximal $t$-ideal of $R$ with $Q \cap H = \emptyset$, then $R_Q$ is a valuation domain. We then use this result to give simple proofs of the facts that (i) $R$ is a UMT-domain if and only if $R_Q$ is a quasi-Pr\"ufer domain for each homogeneous maximal $t$-ideal $Q$ of $R$ and (ii) $R$ is a P$v$MD if and only if every nonzero finitely generated homogeneous ideal of $R$ is $t$-invertible, if and only if $R_Q$ is a valuation domain for all homogeneous maximal $t$-ideals $Q$ of $R$. Let $D[\Gamma]$ be the monoid domain of $\Gamma$ over an integral domain $D$. We also show that $D[\Gamma]$ is a UMT-domain if and only if $D$ is a UMT-domain and the integral closure of $\Gamma_S$ is a valuation monoid for all maximal $t$-ideals $S$ of $\Gamma$. Hence, $D[\Gamma]$ is a P$v$MD if and only if $D$ is a P$v$MD and $\Gamma$ is a P$v$MS.