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.
This is a preview of subscription content, log in via an institution to check access.
References
Birkhoff, G.,Lattice theory, 3rd edition, Amer. Math. Soc. Colloquium Publ., Providence (1967).
Dauns, J. and K. H. Hofmann,Representation of rings by sections, Mem. Amer. Math. Soc. no.83 (1968).
Keimel, K.,Darstellung von Halbgruppen und universellen Algebren durch Schnitte in Garben;bireguläre Halbgruppen, Math. Nachr.45 (1970), 81–96.
Kist, J.,Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. (3)13 (1963), 31–50.
Kist, J.,Compact spaces of minimal prime ideals, Math. Zeitschrift111 (1969), 151–158.
Pierce, R. S.,Modules over commutative rings, Mem. Amer. Math. Soc. no.70 (1967).
Author information
Authors and Affiliations
Additional information
Communicated by K. H. Hofmann
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.
Rights and permissions
About this article
Cite this article
Keimel, K. Baer extensions of rings and stone extensions of semigroups. Semigroup Forum 2, 55–63 (1971). https://doi.org/10.1007/BF02572272
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02572272