 Conversely, it follows from the second, fourth, and last axioms that (-1)"m" is an additive inverse of "m" for all formula_7, so any semimodule over a ring is in fact a module.Any semiring is a left and right semimodule over itself in the same way that a ring is a left and right module over itself.Every commutative monoid is uniquely an formula_8-semimodule in the same way that an abelian group is a formula_9-module.