Algebras of generalized functions with smooth parameter dependence

Abstract: We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\widetilde{\mathbb{K}}{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\widetilde{\mathbb{K}}{sm}$ and establish some properties of its ideals.