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bi-ideals in \(k\)-regular and intra \(k\)-regular semirings. (English) Zbl 1254.16040

Summary: Here we introduce the \(k\)-bi-ideals in semirings and the intra \(k\)-regular semirings. An intra \(k\)-regular semiring \(S\) is a semiring whose additive reduct is a semilattice and for each \(a\in S\) there exists \(x\in S\) such that \(a+xa^2x=xa^2x\). Also it is a semiring in which every \(k\)-ideal is semiprime. Our aim in this article is to characterize both the \(k\)-regular semirings and intra \(k\)-regular semirings using \(k\)-bi-ideals.

MSC:

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