## Semirings which are distributive lattices of $$k$$-simple semirings.(English)Zbl 1274.16066

Summary: Here we characterize the semirings in $$\mathbb{SL}^+$$ that are distributive lattices of $$k$$-simple (left $$k$$-simple) semirings. A semiring is a distributive lattice of $$k$$-simple (left $$k$$-simple) semirings if and only if every $$k$$-ideal (left $$k$$-ideal) is a semiprime $$k$$-ideal. Also a semiring for which every $$k$$-ideal (left $$k$$-ideal) is completely prime is a chain of $$k$$-simple (left $$k$$-simple) semirings.

### MSC:

 16Y60 Semirings 16D25 Ideals in associative algebras 06D05 Structure and representation theory of distributive lattices