  Since for the classification of finite (congruence-)simple semirings it
remains to classify the additively idempotent semirings, we progress on the
characterization of finite simple additively idempotent semirings as semirings
of join-morphisms of a semilattice. We succeed in doing this for many cases,
amongst others for every semiring of this kind with an additively neutral
element. As a consequence we complete the classification of finite simple
semirings with an additively neutral element. To complete the classification of
all finite simple semirings it remains to classify some very specific
semirings, which will be discussed here. Our results employ the theory of
idempotent irreducible semimodules, which we develop further.
