Vol. 29, No. 2, 1969

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Some aspects of Goldie’s torsion theory

Mark Lawrence Teply

Vol. 29 (1969), No. 2, 447–459

Goldie’s torsion class 𝒢 is a class of left R-modules closed under taking submodules, factor modules, extensions, arbitrary direct sums, and injective envelopes. The corresponding Goldie torsionfree class is precisely the class of left R-modules possessing zero singular submodule. It is shown that 𝒢 is closed under taking direct products if and only if nonzero left ideals in have nonzero socles. Another theorem gives four conditions equivalent to the following: Any direct sum of torsionfree injective modules is injective. One of these four conditions is that the ring R is an essential extension of a finite direct sum 𝒢(R) L1 L2 Ln, where each Li is a uniform left ideal of R. It is natural to ask when R actually equals this direct sum. A sufficient condition for this to happen is given. Rings in which every torsionfree principal left ideal is projective are studied. Particular attention is paid to those rings whose Goldie torsion filters possess a cofinal subset of finitely generated left ideals.

Mathematical Subject Classification
Primary: 16.90
Received: 16 February 1968
Published: 1 May 1969
Mark Lawrence Teply