Smooth Modules

Although smooth groups and smooth rings were presented, however the important method that smooth ring acts on smooth group has not been seen in report. In order to solve the problem, this paper defines smooth modulus, smooth homomorphism and smooth quotient modulus by the action. In order to compare two smooth modulus, this paper proves the fundamental theorem of smooth homomorphism of smooth modulus. This paper is different from the classical modulus. The former is based on smooth binary operation and fuzzy equality, and the latter is based on classical binary operation and crisp equality. This paper is different from YUAN fuzzy groups. The former is based on fuzzy equality, and the latter is based on crisp equality. The work of this paper further develops algebraic structures of fuzzy sets.