\(p\)-semisimple modules and type submodules. (English) Zbl 1442.16002
An \(R\)-module \(M\) is \(p\)-semisimple if every submodule is isomorphic to a submodule of a direct summand. Thus \(p\)-semisimplicity is a generalisation of semisimplicity. The authors develop a general theory of \(p\)-semisimplicity with a host of algebraic and topological characterisations, and show that many classes of abelian groups, but not all, are \(p\)-semisimple.
For example, the main theorem states that for a reduced multiplication module \(M\), \(p\)-semisimplicity is equivalent to each of Baer (the annihilator of every submodule is idempotent generated); extending (for every submodule \(N\) there is a summand \(C\) such that \(N\oplus C\) is essential); \(FI\) (every fully invariant submodule is essential in a summand); and Spec\((M)\) is extremally disconnected.
For example, the main theorem states that for a reduced multiplication module \(M\), \(p\)-semisimplicity is equivalent to each of Baer (the annihilator of every submodule is idempotent generated); extending (for every submodule \(N\) there is a summand \(C\) such that \(N\oplus C\) is essential); \(FI\) (every fully invariant submodule is essential in a summand); and Spec\((M)\) is extremally disconnected.
Reviewer: Phillip Schultz (Perth)
MSC:
| 16D10 | General module theory in associative algebras |
Keywords:
\(p\)-semisimple modules; type submodules; multiplication reduced module; Zariski topology; regular closed setsReferences:
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