## On $$k$$-ideals of semirings.(English)Zbl 0779.16020

The paper deals with semirings $$S=(S,+,\cdot)$$ such that $$(S,+)$$ is a commutative inverse semigroup. It contains two results: (1) Each congruence $$\rho$$ on $$S$$ such that $$S\rho$$ is a ring satisfies $$\rho=\{(x,y)\mid x+a_ 1=y+ a_ 2$$ for some $$a_ i\in A\}$$, where $$A$$ is a $$k$$-ideal of $$S$$ containing the set $$E^ +$$ of idempotents of $$(S,+)$$, and conversely. [This was already proved by the same authors in Algebra and number theory, Proc. Symp., Kochi/India 1990, Publ., Cent. Math. Sci., Trivandrum 20, 85-89 (1990), cf. the remarks in the review Zbl 0754.16024.] (2) The $$k$$-ideals $$A$$ of $$S$$ satisfying $$E^ +\subseteq A$$ form a complete modular lattice. [This would be a consequence of Theorem 4.1 in the paper cited above, however, this theorem fails to be true which was overlooked by the referee at that time].

### MSC:

 16Y60 Semirings 16D25 Ideals in associative algebras 20M18 Inverse semigroups 08A30 Subalgebras, congruence relations

Zbl 0754.16024
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