Primitive Ideal Space of $C^{\ast }(R_{+})⋊R^{\times }$

For an integral domain $R$ satisfying certain condition, we characterize the primitive ideal space and its Jacobson topology of the semigroup crossed product $C^{\ast }(R_{+})⋊R^{\times }$. The main example is when $R=\mathbb{Z}[\sqrt{-3}]$.