A note on maximal ideals and prime ideals in ordered semigroups. (English) Zbl 1256.06013
Summary: In [Algebra Discrete Math. 2003, No. 1, 32–35 (2003; Zbl 1035.06003)], N. Kehayopulu et al. showed that in commutative semigroups (resp. ordered semigroups) with identity, every maximal ideal is a prime ideal and the converse is not true in general. In this note we prove that for an ordered semigroup \(S\) satisfying \(S = (S^2]\) all maximal ideals are weakly prime ideals. This generalizes the corresponding result in the above paper. We prove that \(S = (S^2]\) is necessary. Some conditions under which prime ideals are maximal are given.
Citations:
Zbl 1035.06003References:
| [1] | N. Kehayopulu, J. Ponizovskii, and M. Tsingelis, A not on maximal ideals in ordered semigrous, Algebra and DiscreteMathematics 2, 32 (2003). · Zbl 1035.06003 |
| [2] | N. Kehayopulu, On prime, weakly prime ideals in ordered semigrous, Semigroup Forum 44, 32 (1992). · Zbl 0756.06008 |
| [3] | M. Arslanov and N. Kehayopulu, A note on minimal and maximal ideals of ordered semigrous, Lobachevskii J. Math. 11, 3 (2002). · Zbl 1015.06015 |
| [4] | Y. L. Cao and X. Z. Xu, On minimal and maximal left ideals in ordered semigroups, Semigroup Forum 60, 202 (2000). · Zbl 0945.06009 · doi:10.1007/s002339910014 |
| [5] | N. Kehayopulu and M. Tsingelis, On maximal ideals of ordered semigrous, Scientiae Mathematicae Japonicae 55, 61 (2002). · Zbl 1006.06007 |
| [6] | X. Y. Xie and M. F. Wu, SEA Bull. Math. 20, 31 (1996). |
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