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$\mu $-essential submodule; $\mu $-singular module; $\mu $-extending module; weakly $\mu $-extending module
Let $M$ be a module and $\mu $ be a class of modules in $\operatorname{Mod}-R$ which is closed under isomorphisms and submodules. As a generalization of essential submodules Özcan in [8] defines a $\mu $-essential submodule provided it has a non-zero intersection with any non-zero submodule in $\mu $. We define and investigate $\mu $-singular modules. We also introduce $\mu $-extending and weakly $\mu $-extending modules and mainly study weakly $\mu $-extending modules. We give some characterizations of $\mu $-co-H-rings by weakly $\mu $-extending modules. Let $R$ be a right non-$\mu $-singular ring such that all injective modules are non-$\mu $-singular, then $R$ is right $\mu $-co-H-ring if and only if $R$ is a QF-ring.
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