Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-08T00:36:18.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Torsion theories over semihereditary rings

Published online by Cambridge University Press:  09 April 2009

M. W. Evans
M. W. Evans
Affiliation:
Department of Mathematics, St. Michael's Grammar School, Redan Street, St. Kilda, Victoria 3182, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper the class of rings for which the right flat modules form the torsion-free class of a hereditary torsion theory (G, ℱ) are characterized and their structure investigated. These rings are called extended semihereditary rings. It is shown that the class of regular rings with ring homomorphism is a full co-reflective subcategory of the class of extended semihereditary rings with “flat” homomorphisms. A class of prime torsion theories is introduced which determines the torsion theory (G, ℱG). The torsion theory (JG, ℱG) is used to find a suitable generalisation of Dedekind Domain.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 1 , February 1986 , pp. 54 - 70
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Beachy, J., ‘On maximal torsion radicals’, Canad. J. Math. 25 (1973), 712–726.CrossRefGoogle Scholar
[2]Cheatham, T. and Enochs, E., ‘Injective hulls of flat modules’, Comm. Algebra 8 (20) (1980), 1989–1995.CrossRefGoogle Scholar
[3]Dauns, J. and Hofmann, K. H., ‘The representation of biregular rings by sheaves’, Math. Z. 91 (1966), 103–123.CrossRefGoogle Scholar
[4]Eklof, P. and Sabbagh, G., ‘Model-completion and modules’, Ann. Math. Logic 2 (1971), 251–295.CrossRefGoogle Scholar
[5]Evans, M. W., ‘Extensions of semi-hereditary rings’, J. Austral. Math. Soc. (Series A) 23 (1977), 333–339.CrossRefGoogle Scholar
[6]Evans, M. W., ‘Extended semi-hereditary rings’, J. Austral. Math. Soc. (Series A) 26 (1978), 465–474.CrossRefGoogle Scholar
[7]Golan, J. S., Localisation in non-commutative rings (Marcel Dekker Inc., New York, 1974).Google Scholar
[8]Goodearl, K. R., Ring theory (Marcel Dekker Inc., New York, 1976).Google Scholar
[9]Goodearl, K. R., Von Neumann regular rings (Pitman, San Francisco, 1979).Google Scholar
[10]Hattori, A., ‘A foundation of torsion theory for modules over general rings’, Nagoya Math. J. 17 (1960), 147–158.CrossRefGoogle Scholar
[11]Jøndrup, S., ‘On finitely generated flat modules II’, Math. Scand. 27 (1970), 105–112.CrossRefGoogle Scholar
[12]Jøndrup, S., ‘Rings of quotients of some semiprime P.I. rings’. Comm. Algebra 7 (3) (1979), 279–286.CrossRefGoogle Scholar
[13]Lambek, J., ‘Torsion theories, additive semantics and rings of quotients’. Lecture Notes in Mathematics 177, Springer-Verlag, Berlin (1971).Google Scholar
[14]Maths, E., ‘The minimal prime spectrum of a reduced ring’, Illinois J. Math. 27 (1983), 353–391.Google Scholar
[15]Page, A., ‘Sur les anneaux hereditaires ou demi-hereditaires’, Comm. Algebra 6 (11) (1978), 1169–1186.CrossRefGoogle Scholar
[16]Pierce, R. S., ‘Modules over commutative regular rings’, Mem. Amer. Math. Soc. 70 (1967).Google Scholar
[17]Stenström, B., Rings of quotients (Springer-Verlag, Berlin, 1975).CrossRefGoogle Scholar
[18]Storrer, H., ‘Epimorphic extensions of non-commutative rings’, Comment Math. Helt’. 48 (1973), 72–86.CrossRefGoogle Scholar
[19]Turnbridge, D., ‘Torsion theories and rings of quotients of Morita equivalent rings’, Pacific J. Math. 37 (1971), 225–234.CrossRefGoogle Scholar
[20]Vasconcelos, W., ‘Finiteness of projective ideals’, J. Algebra 25 (1973), 269–278.CrossRefGoogle Scholar