Abstract
It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ωω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.
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