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\(\tau\)-supplemented modules and \(\tau\)-weakly supplemented modules. (English) Zbl 1156.16006

Summary: Given a hereditary torsion theory \(\tau=(\mathbb{T,F})\) in Mod-\(R\), a module \(M\) is called \(\tau\)-supplemented if every submodule \(A\) of \(M\) contains a direct summand \(C\) of \(M\) with \(A/C\) \(\tau\)-torsion. A submodule \(V\) of \(M\) is called \(\tau\)-supplement of \(U\) in \(M\) if \(U+V=M\) and \(U\cap V\leq\tau(V)\) and \(M\) is \(\tau\)-weakly supplemented if every submodule of \(M\) has a \(\tau\)-supplement in \(M\). Let \(M\) be a \(\tau\)-weakly supplemented module. Then \(M\) has a decomposition \(M=M_1\oplus M_2\) where \(M_1\) is a semisimple module and \(M_2\) is a module with \(\tau(M_2)\leq_eM_2\). Also, it is shown that any finite sum of \(\tau\)-weakly supplemented modules is a \(\tau\)-weakly supplemented module.

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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