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Abstract
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The notion of extending modules has been generalized to purely extending,
by John Clark. We generalize the notions of C2 and C3, via two ways, by using
purity. We say that a module MR has P.C2 (G.P.C2) property if every submodule
isomorphic to a pure submodule (direct summand) of MR is a pure submodule.
Moreover, we say that a module MR has P.C3 (G.P.C3) property whenever the
sum of two independent pure submodules (direct summands) of MR is a pure
submodule. We study general properties of these modules and investigate the
relations between them.
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