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Filters and overrings

Published online by Cambridge University Press:  09 April 2009

H. H. Brungs
H. H. Brungs
Affiliation:
Dept. Math., University of AlbertaEdmonton, Alberta, Canada
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Let R be an integral domain. It is well known (see Lambek (1971), Stenström (1971)), that idempotent filters of right ideals, torsion radicals and trosio theories are in one-to-one correspondence, but that different idempotent filters F of right ideals may lead to the same rings of quotiens Rf. We have always R ⊃ Rf ⊂Qmax(R). Given this situation one can ask a number of questions. For example: Describe all different idempotent filters for a given ring. Determine all different rings of quotients. When do different filters lead to the same ring of quotients? When are all rings between R and Qmax(R) of the form RF. When is every RF of the form RS−1, where S is an Ore system?

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 4 , June 1975 , pp. 474 - 480
Copyright
Copyright © Australian Mathematical Society 1975

References

Arnold, J. T. & Brewer, J. W. (1971), ‘On flat overrings, ideal transforms and generalized transforms of a commutative ring’, J. Algebra 18, 254–263.CrossRefGoogle Scholar
Beauregard, R. D. (1973), ‘Overrings of Bezout domain’, Can. Math. Bull. 16, 475–477.CrossRefGoogle Scholar
Brungs, H. H. (1971), ‘Overrings of principal ideal domains’, Proc. Amer. Math. Soc. 28, 44–46.CrossRefGoogle Scholar
Brungs, H. H. (1973), ‘Non commutative Krull domains’, J. Reine Angew. Math. 264, 161–171.Google Scholar
Gilmer, R. W. & Ohm, J. (1964), ‘Integral domains with quotient overrings’, Math. Ann. 153, 97–103.CrossRefGoogle Scholar
Henriksen, M. (1952), ‘On the ideal structure of the ring of entire functions’, Pac. J. Math. 2, 179–184.CrossRefGoogle Scholar
Lambek, J. (1971), Torsion theories, additive semantics and rings of quotients Lecture Notes in Mathematics No. 177, (Springer-Verlag, New York, 1971).CrossRefGoogle Scholar
Popescu, N. & Spirzu, T. (1970), ‘Quelques observations sur les épimorphismes plats (à gauche) d'anneaux’, J. Algebra 16, 40–59.CrossRefGoogle Scholar
Stenström, B., Rings and modules of quotients. Lecture Notes in Mathematics, No. 237. (Springer-Verlag, New York, 1971).CrossRefGoogle Scholar
Turnidge, D. R. (1970), ‘Torsion theories and semihereditary rings’. Proc. Amer. Math. Soc. 23, 137–143.CrossRefGoogle Scholar