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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
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