  In this paper, we introduce and study V- and CI-semirings---semirings all of
whose simple and cyclic, respectively, semimodules are injective. We describe
V-semirings for some classes of semirings and establish some fundamental
properties of V-semirings. We show that all Jacobson-semisimple V-semirings are
V-rings. We also completely describe the bounded distributive lattices,
Gelfand, subtractive, semisimple, and anti-bounded, semirings that are
CI-semirings. Applying these results, we give complete characterizations of
congruence-simple subtractive and congruence-simple anti-bounded CI-semirings
which solve two earlier open problems for these classes of CI-semirings.
