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 . In this paper, 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. Our results show that there is also a class of weakly Krull domains, which are not Krull but have full system of sets of lengths.