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Neat submodules by relative height. (English) Zbl 0683.16023
Let R be a ring with unity element, and let \(\sigma\) be a hereditary torsion theory of left R-modules. Some \(\sigma\)-torsionfree modules M that behave like abelian groups are studied. To accomplish this study, the author assumes the conditions: (I) every \(\sigma\)-finitely generated submodule of a \(\sigma\)-torsionfree homomorphic image of M is a direct sum of strongly \(\sigma\)-uniserial modules; and (II) if X is a \(\sigma\)- torsionfree homomorphic image of M, then for any \(\sigma\)-uniserial submodule U of X, any \(\sigma\)-closed \(\sigma\)-uniserial submodule V of X, and any submodule W of U, each homomorphism f: \(W\to V\) can be extended to g: \(U\to V\) whenever \(U/C\ell^ U_{\sigma}(W)\) has shorter \(\sigma\)-composition length than \(V/C\ell^ V_{\sigma}(f(W))\). Neat and closed submodules of M are studied by using the properties of height relative to \(\sigma\).
Reviewer: M.L.Teply

MSC:
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16D80 Other classes of modules and ideals in associative algebras
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