Smooth rings

Different from available Rosenfeld fuzzy groups and YUAN fuzzy groups, smooth groups are a new kind of fuzzy group structure. However, smooth groups possess one kind of binary operations. In order to solve the problem, this paper forwards smooth rings which possess two kinds of binary operations, presents and proves two theorems: 1. smooth kernel of smooth homomorphism sigma is two-sided ideal. The sufficient and necessary condition for sigma(r₁) = sigma(r₂) is that r₁ and r₂ are congruent for smooth kernel. 2. Fundamental theorem of homomorphism of smooth rings. The work of this paper enriches the algebraic structure of smooth groups. Similar work has not been seen in literature.