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A relative Ulm’s theorem. (English) Zbl 0779.16013
Let $$R$$ be an associative ring with 1, and let $$\sigma$$ be a hereditary torsion theory for $$R$$-mod. For a submodule $$N$$ of $$M$$, we let $$C\ell^ M(N)$$ be the $$\sigma$$-closure of $$N$$ in $$M$$, and we let $$\ell(M)$$ denote the $$\sigma$$-composition length of $$M$$ when it exists. A $$\sigma$$-torsionfree module $$M$$ is called $$\sigma$$-uniserial if it has exactly one $$\sigma$$-composition series. Let $$H^ M(x) = \sup\{\ell(N/C\ell^ N(Rx)\mid N$$ is a $$\sigma$$-uniserial submodule of $$M$$ containing $$x$$}, and let $$H_ k(M) = C\ell^ M(\sum\{Rx\mid x$$ is a uniform element of $$M$$ with $$H^ M(x) \geq k$$}). Then $$M$$ is called $$h$$-divisible if $$M = \cap H_ k(M)$$. All the results of the paper are about $$\sigma$$- torsionfree modules that satisfy the following two conditions: (I) Every $$\sigma$$-finitely generated submodule of a $$\sigma$$-torsionfree homomorphic image of $$M$$ is a direct sum of $$\sigma$$-uniserial modules, and (II) if $$X$$ is a $$\sigma$$-torsionfree homomorphic image of $$M$$, then for any $$\sigma$$-uniserial $$N\subseteq M$$, any $$\sigma$$-closed $$\sigma$$- uniserial $$V$$ of $$X$$, and submodule $$W$$ of $$N$$, there is an extension $$g: N\to V$$ of each homomorphism $$f: W\to V$$, provided that $$\ell[N/C\ell^ N(W)] \leq \ell[V/C\ell^ V(f(W))]$$. A module $$M$$ satisfying (I) and (II) is $$h$$-divisible if and only if $$M$$ is a direct sum of uniform $$h$$- divisible modules. A submodule $$B$$ of $$M$$ is called basic if $$B$$ is a direct sum of $$\sigma$$-closed $$\sigma$$-uniserial modules, $$B$$ is $$h$$-pure in $$M$$, and $$M/B$$ is $$h$$-divisible. Every module $$M$$ satisfying (I) and (II) has a basic submodule, and any two basic submodules of $$M$$ are isomorphic. Let $$M$$ and $$N$$ be $$\sigma$$-countably generated $$\sigma$$- torsionfree modules such that $$M\oplus N$$ satisfies conditions (I) and (II); then $$M$$ and $$N$$ are isomorphic if and only if they have isomorphic Ulm factors (which are defined analogously to Ulm factors for Abelian groups).
##### MSC:
 16S90 Torsion theories; radicals on module categories (associative algebraic aspects) 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 20K99 Abelian groups