In this paper, we study the class group structure of rings of the form \(D + E[\![\Gamma ^*]\!] \) , where \(D \subseteq E\) is an extension of integral domains and \(\Gamma \) is a numerical semigroup. We examine the properties of t-invertible and v-invertible fractional ideals in these rings and investigate their relationship with the t-class group \(\operatorname {Cl}_t(D)\) of the base domain D. Our primary results establish that the natural mapping \(\varphi : \operatorname {Cl}_t(D) \rightarrow \operatorname {Cl}_t(D + E[\![\Gamma ^*]\!] )\) is an injective homomorphism when E is a flat D-module, although it is not surjective in general. Furthermore, we provide a complete characterization of the t-invertible t-ideals of \(D + E[\![\Gamma ^*]\!] \) extended (with nonzero trace) to D. Specifically, we show that if E is completely integrally closed and \(\operatorname {qf}(D) \subseteq E,\) then every t-invertible t-ideal I of \(D+E[\![\Gamma ^*]\!] \) with nonzero trace in D can be expressed as \(I = uJ(D+E[\![\Gamma ^*]\!] ),\) for some \(u\in \operatorname {qf}(D+E[\![\Gamma ^*]\!] ),\) and a nonzero t-ideal J of D.
