Modules satisfying both chain conditions with respect to a torsion theory

Golan, Jonathan S.

doi:10.1090/S0002-9939-1975-0429981-2

Modules satisfying both chain conditions with respect to a torsion theory
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by Jonathan S. Golan

Proc. Amer. Math. Soc. 52 (1975), 103-108

DOI: https://doi.org/10.1090/S0002-9939-1975-0429981-2

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Abstract:

Goldman [3] has introduced the notion of the length of a module with respect to a torsion theory and has studied finitely-generated modules over left noetherian rings which have finite length. In this note we simplify the proofs of some of Goldman’s results and generalize them by removing both the finite-generation and noetherianness conditions.

References

Jonathan S. Golan, Topologies on the torsion-theoretic spectrum of a noncommutative ring, Pacific J. Math. 51 (1974), 439–450. MR 369439
Oscar Goldman, Rings and modules of quotients, J. Algebra 13 (1969), 10–47. MR 245608, DOI 10.1016/0021-8693(69)90004-0

Elements of non-commutative arithmetic

Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. With an appendix by H. H. Storrer on torsion theories and dominant dimensions. MR 0284459

Finiteness conditions and torsion theories

J. Raynaud, Localisations et anneaux semi-noéthériens à droite, Publ. Dép. Math. (Lyon) 8 (1971), no. 3-4, 77–112 (French). MR 335561
Bo Stenström, Rings and modules of quotients, Lecture Notes in Mathematics, Vol. 237, Springer-Verlag, Berlin-New York, 1971. MR 0325663
Hans H. Storrer, On Goldman’s primary decomposition, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970–1971, Vol. I), Lecture Notes in Math., Vol. 246, Springer, Berlin, 1972, pp. 617–661. MR 0360717