On some Regularity in Semirings

We prove the following theorem: Let R be a right k-semiring whose every simple singular right R-semimodule is p-injective. Then (1) Every large right ideal of R is idempotent. (2) If R has no nonzero nilpotent right ideals, then R is fully right idempotent. (3) If R is right duo, then R is right π-regular.