On the Socles of Commutator Invariant Submodules of $QTAG$-Modules

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In [9, 8, 10], respectively, socles have been studied with the aid of fully invariant, characteristic and projection invariant submodules of $QTAG$-modules. Here we focus our attention on the socles of commutator invariant submodules and introduce a new class of modules, which we term commutator socle-regular $QTAG$-modules. After establishing some crucial properties of commutator socle-regularity, we show that the addition of separable summand to a module does not influence commutator socle-regularity.