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QTAG torsionfree modules. (English) Zbl 0765.16009
This paper extends the results of S. Singh [Acta Math. Hung. 50, 85-95 (1987; Zbl 0628.16014)] and the authors on modules that behave like abelian groups. The results are obtained in terms of a hereditary torsion theory \(\sigma\). A \(\sigma\)-torsionfree module \(M\) is called a \(\sigma\)- QTAG-module if \(M\) satisfies the 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) for every \(\sigma\)- uniserial submodule \(A\) of a \(\sigma\)-torsionfree homomorphic image \(N\) of \(M\), every homomorphism \(f: A\to K\) into a \(\sigma\)-closed, \(\sigma\)- uniserial submodule \(K\) of \(N\) can be extended to a homomorphism \(g: C\ell(A)\to K\). The properties of \(\sigma\)-QTAG-modules are investigated in detail. The results include theorems on closed submodules of \(\sigma\)- QTAG-modules, decompositions, \(h\)-purity (defined in terms of the \(\sigma\)-height of a uniform element), and \(h\)-neatness.
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)
16D80 Other classes of modules and ideals in associative algebras
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