Abstract: As a proper generalization of factor-square-full modules, the concept of factor-square-full semimodules is introduced in this paper. Let S be a semiring and M a left S-semimodule. A proper subtractive subsemimodule X of M is said to be d-square if there exists a normal epimorphism f:M→(M/X)²; A proper subtractive subsemimodule X of M is said to be quasi-d-square if there exists a d-square subsemimodule Y such that X≤Y;M is said to factor-square-full if any proper subsemimodule is quasi-d-square. The properties of these subsemimodule classes and factor-square-full semimodules are discussed.

Key words: normal epimorphism, d-square subsemimodules, quasi-d-square subsemimodules, factor-square-full semimodules