$V$-semirings

We investigate the semirings over which all simple semimodules are injective. In ring and module theory, the rings with an analogous condition are called $V$-rings. Therefore it is natural to call the semirings under consideration $V$-semirings. We obtain the semiring analogs of some well-known results on $V$-rings, including an analog of Kaplansky's theorem on commutative $V$-rings.