Semigroup rings as weakly Krull domains

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