\(\aleph _ 0\)-continuous modules (Q1066985)

A module M is called \(\aleph_ 0\)-continuous if every countably- generated submodule K is essential in some direct summand N of M, and any submodule of M isomorphic to N is also a direct summand of M. The author studies non-singular \(\aleph_ 0\)-continuous modules over regular (in the sense of von Neumann) rings, extending a result of Goodearl. He shows that, for a non-singular \(\aleph_ 0\)-continuous module M over a regular ring, M has no proper direct summand isomorphic to M if and only if \(E_{{\mathcal F}}(M)\) has the same property, where \(E_{{\mathcal F}}(M)\) is the submodule of M consisting of all elements in the injective hull of M which are sent into M by some countably generated essential right ideal of R.