Semigroup rings as weakly Krull domains

Let $D$ be an integral domain and $\Gamma$ a torsion-free commutative cancellative (additive) semigroup with identity element quotient group $G$. In this paper, we show that if char$(D)=0$ (resp., char$(D)=p>0$), then $D[\Gamma]$ is weakly Krull only UMT-domain, UMT-monoid, $G$ of type $(0,0,0, \dots )$ except $p$). Moreover, give arithmetical applications result.