Geodesic
A note on maximal ideals in ordered semigroups
Algebra and discrete mathematics, no. 1 (2003), pp. 32-35 Cet article a éte moissonné depuis la source Math-Net.Ru

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In commutative rings having an identity element, every maximal ideal is a prime ideal, but the converse statement does not hold, in general. According to the present note, similar results for ordered semigroups and semigroups-without order-also hold. In fact, we prove that in commutative ordered semigroups with identity each maximal ideal is a prime ideal, the converse statement does not hold, in general.
Keywords: maximal ideal, prime ideal in ordered semigroups. 
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N. Kehayopulu; J. Ponizovskii; M. Tsingelis. A note on maximal ideals in ordered semigroups. Algebra and discrete mathematics, no. 1 (2003), pp. 32-35. http://geodesic.mathdoc.fr/item/ADM_2003_1_a3/