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Quasi \(k\)-ideals in \(k\)-regular and intra \(k\)-regular semirings. (English) Zbl 1249.16053

Summary: A semiring \(S\) whose additive reduct is a semilattice, is called an intra \(k\)-regular semiring if for each \(a\in S\) there exists \(x\in S\) such that \(a+xa^2x=xa^2x\) and is called a \(k\)-regular semiring if for each \(a\in S\) there exists \(x\in S\) such that \(a+axa=axa\). Here we introduce quasi \(k\)-ideals in semirings and characterize both the \(k\)-regular and intra \(k\)-regular semirings by their quasi \(k\)-ideals.

MSC:

16Y60 Semirings
16D25 Ideals in associative algebras
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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