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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.

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
Full Text: DOI
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