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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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