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

Bhuniya, A. K.; Mondal, T. K.

Southeast Asian Bull. Math. 36, No. 3, 309-318 (2012).

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

Keywords:

simple semirings; semiprime ideals; completely prime ideals; distributive lattices