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Representation and extension of semiprime rings

Published online by Cambridge University Press:  09 April 2009

G. Davis
G. Davis
Affiliation:
Department of Mathematics La Trobe University Bundoora, Vic. 3083, Australia
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In many respects the theory of semi-prime rings (i.e. rings without proper nilpotents) is similar to that for lattice-ordered groups. In this paper semi-prime rings are faithfully represented as subrings of continuous global sections of sheaves of integral domains with Boolean base spaces. This representation allows a simple description of a particular extension of a semi-prime ring as the corresponding ring of all continuous global sections. The ideals in a semi-prime ring R that give rise to the stalks in the sheaf representation are then characterized when R is projectable. Finally equivalent conditions are given for a semi-prime ring R to satisfy a condition, that in the case of lattice-groups, was termed “weak projectability” by Spirason and Strzelecki [8]. Some of the results that are common to semi-prime rings and lattice-groups (and semi-prime semigroups) have been extended to certain universal algebras by Davey [3].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 3 , May 1973 , pp. 353 - 365
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Adams, D. H., ‘Semigroups with no non-zero nilpotents’, Math Z. 123 (1971), 168–176.CrossRefGoogle Scholar
[2]Dauns, J. and Hofmann, K. H., ‘The representation of biregular rings by sheaves’, Math. Z. 91 (1966), 103–123.CrossRefGoogle Scholar
[3]Davey, B., Sheaves of universal algebrvs. (M. Sc. Thesis, Monash University, 1972).Google Scholar
[4]Keimel, K., ‘Baer extensions of rings and Stone extensions of semigroups’, Semigroup Forum Vol. 2 (1971), 55–63.CrossRefGoogle Scholar
[5]Kist, J., ‘Compact spaces of minimal prime ideals’, Math. Z. 111 (1969), 151–158.CrossRefGoogle Scholar
[6]Koh, K., ‘On functional representations of a ring without nilpotent elements’, Canad. Math. Bull. 14 (3) (1971), 349–352.CrossRefGoogle Scholar
[7]Spirason, G. T. and Strzelecki, E., ‘A note on P t-ideals’, J. of Aust. Math. Soc. 14 (1972), 304–310.CrossRefGoogle Scholar
[8]Veksler, A. I., ‘Linear spaces with disjoint elements and their conversion into vector lattices’, Leningrad Gas. Ped. Inst. Ucen. Zap. 328 (1967), 19–43.Google Scholar