In this thesis, several characterizations of certain rings via FC-purity and I-purity are considered. In particular, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Kothe ring. Also, we introduce the notion of (m,n)-algebraically compact modules as an analogue of algebraically compact modules and then we show that (m,n)-algebraic compactness and (m,n)-pure injectivity for modules coincide. Finally, we tudy commutative local rings for which every ideal is a direct sum of cyclic modules.
