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Semigroups containing no maximal ideals. (English) Zbl 0685.20045

A two-sided ideal \(M\) of a semigroup \(S\) is said to be covered if \(M\subseteq S(S\setminus M)S\). The author proves that a semigroup has no maximal ideals iff each of its principal ideals is covered. If a semigroup has no greatest ideal and the set of its principal ideals is up-directed then it has no maximal ideals. Similar results hold for maximal one-sided ideals.
Reviewer: L.Márki

MSC:

20M12 Ideal theory for semigroups
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References:

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