  Flat modules play an important role in the study of the category of modules
over rings and in the characterization of some classes of rings. We study the
e-flatness for semimodules introduced by the first author using his new notion
of exact sequences of semimodules and its relationships with other notions of
flatness for semimodules over semirings. We also prove that a subtractive
semiring over which every right (left) semimodule is e-flat is a von Neumann
regular semiring.
