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