Absolute integrability and its absolute value inequality for fuzzy-number-valued functions are worth to be considered. In this paper, absolute integrability and its absolute valueine quality for fuzzy-number-valued functions are discussed by means of the characteristic theorems of nonabsolute fuzzy integrals and the embedding theorem, i.e., the fuzzy number space can be embedded into a concrete Banach space. Several necessary and sufficient conditions and examples are given.