##
**\(\oplus\)-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.

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 |