Baer extensions of rings and stone extensions of semigroups |
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Authors: | Klaus Keimel |
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Affiliation: | (1) Collège Scientifique Universitaire, Parc Grand-Mont 37, Tours, (France) |
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Abstract: | Let R be a commutative semigroup [resp. ring] with identity and zero, but without nilpotent elements. We say that R is a Stone semigroup [Baer ring], if for each annihilator ideal P⊂R there are idempotents e1 ε P and e2 ε Ann(P) such that x→(e1x, e2x):R→P×Ann(P) is an isomorphism. We show that for a given R there exists a Stone semigroup [Baer ring] S containing R that is minimal with respect to this property. In the ring case, S is uniquely determined if one requires that there be a natural bijection between the sets of annihilator ideals of R and S. This is close to results of J. Kist [5]. Like Kist, we use elementary sheaf-theoretical methods (see [2], [3], [6]). Proofs are not very detailed. An address delivered at the Symposium on Semigroups and the Multiplicative Structure of Rings, University of Puerto Rico, Mayaguez, Puerto Rico, March 9–13, 1970. |
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