Geodesic
Abelian Groups in Which Every α-Pure Subgroup is β-Pure
Canadian journal of mathematics, Tome 25 (1973) no. 3, pp. 560-566

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The determination of the abelian groups in which every neat subgroup is pure is a relatively routine exercise (see [6]). There are numerous problems of this type; for example, the determination of the groups in which every pure subgroup is isotype or the groups in which every subgroup is isotype. These are all special cases of the general problem of determining the abelian groups in which every α-pure subgroup is β-pure for arbitrary ordinal numbers α and β. The solution of this general problem is the object of this paper. 
Moore, J. Douglas; Hewett, Edwin J. Abelian Groups in Which Every α-Pure Subgroup is β-Pure. Canadian journal of mathematics, Tome 25 (1973) no. 3, pp. 560-566. doi: 10.4153/CJM-1973-057-9
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