Completely $0$-simple semirings

A completely $( - 0)$ simple semiring is a semiring $R$ which is $( - 0)$ simple and is the union of its $( - 0)$ minimal left ideals and the union of its $( - 0)$ minimal right ideals. Structure results are obtained for such semirings. First the multiplicative semigroup of $R$ is completely $( - 0)$ simple; for any $\mathcal {H}$-class $H( \ne 0),H( \cup \{ 0\} )$ is a subsemiring. If furthermore $R$ has a zero but is not a division ring, and if $(H \cup \{ 0\} , + )$ has a completely simple kernel for some $H$ as above (for instance, if $R$ is compact or if the $\mathcal {H}$-classes are finite), then (i) $(R, + )$ is idempotent; (ii) $R$ has no zero divisors, additively or multiplicatively. Additional results are given, concerning the additive $\mathcal {J}$-classes of $R$ and also $( - 0)$ minimal ideals of semirings in general.