Abstract
We shall consider semigroups with O, which contain a unique maximal right ideal generated by a finite number of independent generators and in which every proper right ideal is contained in the unique maximal right ideal and investigate when these semigroups are multiplicative semigroups of a ring. We prove in particular that the necessary condition for this class of semigroups S to admit ring structure is S=S2 if |S|>2. Furthermore the admissible ring structure of S is determined when the product of every two generators of the maximal right ideal M is O and when S satisfies one of the two conditions, namely S is commutative without idempotents except O and 1 or every generator of M is nilpotent.
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References
Cartan, H. and S. Eilenberg,Homological Algebra, Princeton, N.J., 1956.
Satyanarayana, M.,On semigroups admitting ring structure, Semigroup Forum 3(1971), 43–50.
—,Principal right ideal semigroups, J. London Math. Soc. (2), 3(1971), 549–553.
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Communicated by M. Petrich
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Satyanarayana, M. On semigroups admitting ring structure II. Semigroup Forum 6, 189–197 (1973). https://doi.org/10.1007/BF02389122
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DOI: https://doi.org/10.1007/BF02389122