It can be shown that if $R$ is a (not necessarily commutative) ring, then all left $R$-modules are free iff $R$ is a division ring (Every $R$-module is free $\implies$ $R$ is a division ring). Is there an analogous result for semirings? For instance is it true that if $R$ is a (not necessarily commutative) semiring, then all left $R$-semimodules are free iff $R$ is a division ring?
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