A note on flat semimodules and von Neumann regular semirings

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 [Formula: see text]-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 [Formula: see text]-flat is a von Neumann regular semiring.