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\(\oplus\)-supplemented modules relative to a torsion theory. (English) Zbl 1104.16026

Let \(R\) be a ring with identity and \(M\) a right \(R\)-module. Given a hereditary torsion theory \(\tau\) in mod-\(R\), \(M\) is called \(\tau\)-\(\oplus\)-supplemented if for every submodule \(N\) of \(M\) there exists a direct summand \(K\) of \(M\) such that \(M=N+K\) and \(N\cap K\) is \(\tau\)-torsion. \(M\) is called completely \(\tau\)-\(\oplus\)-supplemented if every direct summand of \(M\) is \(\tau\)-\(\oplus\)-supplemented. Also \(M\) is called strongly \(\tau\)-\(\oplus\)-supplemented if for every submodule \(N\) of \(M\) there exists a direct summand \(K\) of \(M\) such that \(M=N+K\) and \(N\cap K\) is a small \(\tau\)-torsion submodule of \(M\).
Then the authors give here some fundamental properties of these classes of modules and study the decomposition of \(\tau\)-\(\oplus\)-supplemented modules under certain conditions on modules. The question of which direct sums of \(\tau\)-\(\oplus\)-supplemented modules are \(\tau\)-\(\oplus\)-supplemented is treated here.
An epimorphism \(f\colon P\to M\) is called a \(\tau\)-projective cover of \(M\) if \(P\) is \(\tau\)-projective and \(\text{Ker}(f)\) is a small \(\tau\)-torsion submodule of \(P\). The ring \(R\) is called right \(\tau\)-perfect if every right \(R\)-module has a \(\tau\)-projective cover. It is then shown that \(R\) is right \(\tau\)-perfect if and only if every projective \(R\)-module is strongly \(\tau\)-\(\oplus\)-supplemented.
Finally the authors give an example of a torison theory \(\tau\) and a module \(M\) relative to which \(M\) is \(\tau\)-\(\oplus\)-supplemented but not \(\tau\)-supplemented.
Reviewer: Y. Kurata (Hadano)

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)
16D40 Free, projective, and flat modules and ideals in associative algebras
16L30 Noncommutative local and semilocal rings, perfect rings
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