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When is the CS condition hereditary? (English) Zbl 0946.16004
A module \(M\) is said to be CS if every submodule of \(M\) is essential in a direct summand of \(M\). Every injective module is CS, and so it is far from being true that every submodule of a CS module is CS. The authors investigate some situations in which it can be shown that certain types of submodules of a CS module are CS. For instance every fully invariant submodule of a CS module is CS. Let \(R\) be a ring. If \(R\) is right CS and every idempotent element of \(R\) is central, then every right ideal of \(R\) is CS (as a right \(R\)-module). If \(R\) is right non-singular and right CS then every principal right ideal of \(R\) is CS, but an example is given to show that this does not extend to all finitely-generated right ideals.

MSC:
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D25 Ideals in associative algebras
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