## 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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