Fabrici, Imrich Semigroups containing no maximal ideals. (English) Zbl 0685.20045 Semigroup Forum 40, No. 1, 101-104 (1990). 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 Cited in 1 Review MSC: 20M12 Ideal theory for semigroups Keywords:covered ideals; maximal ideals; principal ideals PDFBibTeX XMLCite \textit{I. Fabrici}, Semigroup Forum 40, No. 1, 101--104 (1990; Zbl 0685.20045) Full Text: DOI EuDML References: [1] Fabrici, I. and T. Macko,On bases and maximal ideals in semigroups, Math. Slovaca 31 (1981) 115–120. · Zbl 0461.20045 [2] Fabrici, I.,Semigroups containing covered one-sided ideals, Math. Slovaca 31 (1981) 225–231. · Zbl 0465.20061 [3] Fabrici, I.,Semigroups containing covered two-sided ideals, Math. Slovaca 34 (1984) 355–363. · Zbl 0601.20056 [4] Clifford, A.H. and G.B. Preston,The algebraic theory of semigroups, American Math. Soc., Providence, R.I. 1961. · Zbl 0111.03403 [5] Howie, J.M.,An introduction to semigroup theory, Academic Press, 1976. · Zbl 0355.20056 [6] Schwarz, Š.,Prime ideals and maximal ideals in semigroups, Czechoslov. Math. J. 19 (1966) 72–79. · Zbl 0176.29503 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.