Smooth Projective Modules

Aimed at that available application of smooth homomorphism is around the comparison of two smooth modulus with each other and any of its application to construct new smooth modulus has not been seen, this paper applies smooth homomorphism and modulus to construct smooth projective modulus and proves three theorems:1. the theorem that lifts some examples of smooth projective modulus. 2. the sufficient and necessary condition that the direct sum of two smooth modulus is smooth projective modulus; 3. the uniqueness of smooth projective module. The smooth projective is different from classical module. The former is based on fuzzy equality and smooth binary operation, the latter is based on crisp equality and classical binary operation. This paper is different from the fundamental theorem of smooth homomorphism of modulus in application of homomorphism . The former applies homomorphism to construct new module--projective module, the latter applied homomorphism to comparison of two smooth modulus with each other.