On purity relative to an hereditary torsion theory.
Gray, Derek Johanathan.
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The thesis is mainly concerned with properties of the concept "σ-purity" introduced by J. Lambek in "Torsion Theories, Additive Semantics and Rings of Quotients", (Springer-Verlag, 1971). In particular we are interested in modul es M for which every exact sequence of the form O→M→K→L→O (or O→K→M→L→O or O→K→L→M→O) is σ-pure exact. Modules of the first type turn out to be precisely the σ- injective modules of O. Goldman (J. Algebra 13, (1969), 10-47). This characterization allows us to study σ- injectivity from the perspective of purity. Similarly the demand that every short exact sequence of modules of the form O→K→M→L→O or O→K→L→M→O be σ-pure exact leads to concepts which generalize regularity and flatness respectively. The questions of which properties of regularity and flatness extend to these more general concepts of σ- regularity and σ-flatness are investigated. For various classes of rings R and torsion radicals σ on R-mod, certain conditions equivalent to the σ-regularity and the σ-injectivity of R are found. We also introduce some new dimensions and study semi-σ-flat and semi-σ-injective modules (defined by suitably restricting conditions on σ-flat and σ-injective modules). We further characterize those rings R for which every R-module is semi- σ-flat. The related concepts of a projective cover and a perfect ring (introduced by H. Bass in Trans. Amer. Math. Soc. 95, (1960), 466-488) are extended in a 'natural way and, inter alia , we obtain a generalization of a famous theorem of Bass. Lastly, we develop a relativized version of the Jacobson Radical which is shown to have properties analogous to both the classical Jacobson Radical and a radical due to J.S. Golan.