On injectivity of semimodules over additively idempotent division semirings and chain MV-semirings

In this paper, we give a characterization of injective semimodules over additively idempotent semirings. Consequently, we provide a complete description of injective semimodules over the semifield of tropical integers, and give an explicit construction of the injective hulls of semimodules over chain division semirings. We also give a criterion for self-injective MV-semirings with an atomic Boolean center, and describe the structure of (finitely generated) injective semimodules over finite MV-semirings, as well as we show that every complete MV-semiring with an atomic Boolean center is an exact semiring which is defined by a Hahn-Banach-type separation property on semimodules arising in the tropical case from the phenomenon of tropical matrix duality. Moreover, we show that complete Boolean algebras are precisely the MV-semirings in which every principal ideal is injective.