Monomial algebraic and algebroid curves and their Lipschitz saturation over the ground ring

In this article, we provide an explicit description of the Lipschitz saturation $A_{B,R}^{\ast }$ of a subalgebra $A\subseteq R[[t^{{\gamma }_1},\dots ,t^{{\gamma }_n}]]$ over a ring $R$ in terms of the numerical semigroup $\mathrm{\Gamma }=⟨{\gamma }_1,\dots ,{\gamma }_n⟩$ with mild conditions imposed on $R$.