Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-12-01T03:58:09.711Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice-ordered modules of quotients

Part of: Ordered structures

Published online by Cambridge University Press:  09 April 2009

Stuart A. Steinberg
Stuart A. Steinberg
Affiliation:
University of Toledo, Toledo, Ohio 43606, U.S.A.
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.

Let Q be the ring of quotients of the f-ring R with respect to a positive hereditary torsion theory and suppose Q is a right f-ring. It is shown that if the finitely-generated right ideals of R are principal, then Q is an f-ring. Also, if QR is injective, Q is an f-ring if and only if its Jacobson radical is convex. Moreover, a class of po-rings is introduced (which includes the classes of commutative po-rings and right convex f-rings) over which Q(M) is an f-module for each f-module M.

MSC classification

Secondary: 06F25: Ordered rings, algebras, modules
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 2 , December 1980 , pp. 243 - 251
Copyright
Copyright © Australian Mathematical Society 1980

References

Anderson, F. W. (1965), ‘Lattice-ordered rings of quotients’, Canad. J. Math. 17, 434–448.Google Scholar
Bigard, A. (1973), ‘Theories de torsion et f-modules’, Seminaire P. Dubreil (Algebre, Exp. No. 5. Secretariat Mathematique, Paris).Google Scholar
Birkhoff, G. and Pierce, R. S. (1956), ‘Lattice-ordered rings’, An. Acad. Brasil. Ci. 28, 41–69.Google Scholar
Conrad, P. (1970), Lattice-ordered groups (Tulane University, New Orleans, La., U.S.A.)Google Scholar
Diem, J. E. (1968), ‘A radical for lattice-ordered rings’, Pacific J. Math. 25, 71–82.Google Scholar
Georgoudis, J. (1972), Torsion theories and f-rings, Ph.D. thesis (McGill University, Montreal, Quebec).Google Scholar
Goldman, O. (1969), ‘Rings and modules of quotients’, J. Algebra 13, 10–47.Google Scholar
Henriksen, M. (1977), ‘Semiprime ideals of f-rings’, Symposia Mathematica, 21, 401–409.Google Scholar
Johnson, D. G. and Kist, J. (1962), ‘Prime ideals in vector lattices’, Canad. J. Math. 14, 517–528.Google Scholar
Lambek, J. (1966), Lectures on rings and modules (Blaisdell, Waltham, Ma., U.S.A.).Google Scholar
Steinberg, S. A. (1972a), ‘Finitely-valued f-modules’, Pacific J. Math. 40, 723–737.Google Scholar
Steinberg, S. A. (1972b), ‘An embedding theorem for commutative lattice-ordered domains’, Proc. Amer. Math. Soc. 31, 409–416.CrossRefGoogle Scholar
Steinberg, S. A. (1972c), ‘Lattice-ordered injective hulls’, Trans. Amer. Math. Soc. 169, 365–388.Google Scholar
Steinberg, S. A. (1973), ‘Quotient rings of a class of lattice-ordered rings’, Canad. J. Math. 25, 627–645.CrossRefGoogle Scholar
Steinberg, S. A. (1976), ‘On lattice-ordered rings in which the square of every element is positive’, J. Austral. Math. Soc. Ser. A 22, 362–370.Google Scholar
Steinberg, S. A. (1978), ‘Lattice-ordered modules of quotients’, Notices Amer. Math. Soc. 25, No. 2, p. A–271.Google Scholar
Stenstron, B. (1971), Rings and modules of quotients (Lecture notes in Math. 237, Springer-Verlag, Heidelberg).Google Scholar
Zaharoff, V. K. (1977), ‘Divisible hull and orthocompletion of lattice-ordered modules’, (in Russian), Mat. Sb. 103 (145), 346–357.Google Scholar