Some properties of purity in Abelian groups

Pure submodules of an Abelian group as modules over its endomorphism ring are studied. In the first section of the paper the endopure submodules of a quasi-decomposable torsion free Abelian group of rank 3 are described. In the second section we prove that pure injectivity of an Abelian group over its endomorphism ring is equivalent to its algebraic compactness. This result is generalized for unitary modules over associative rings with unity.