Simple semirings with a bi-absorbing element

We study additively idempotent congruence-simple semirings with a bi-absorbing element. We characterize a subclass of these semirings in terms of semimodules of a special type (o-characteristic semimodules). We show that o-characteristic semimodules are uniquely determined. We also generalize a result by Ježek and Kepka on simple semirings of endomorphisms of semilattices.