Abstract
A right-chain semigroup is a semigroup whose right ideals are totally ordered by set inclusion. The main result of this paper says that if S is a right-chain semigroup admitting a ring structure, then either S is a null semigroup with two elements or sS=S for some s∈S. Using this we give an elementary proof of Oman’s characterization of semigroups admitting a ring structure whose subsemigroups (containing zero) form a chain. We also apply this result, along with two other results proved in this paper, to show that no nontrivial multiplicative bounded interval semigroup on the real line ℝ admits a ring structure, obtaining the main results of Kemprasit et al. (ScienceAsia 36: 85–88, 2010).
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Communicated by Michael W. Mislove.
Dedicated to Professor Jan Skowroński.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Mazurek, R. On semigroups admitting ring structure. Semigroup Forum 83, 335–342 (2011). https://doi.org/10.1007/s00233-011-9316-8
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DOI: https://doi.org/10.1007/s00233-011-9316-8