In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand.
It has found many uses in abelian group theory and related areas.
The work of Prüfer was complemented by Kulikoff[3] where many results were proved again using pure subgroups systematically.
In particular, a proof was given that pure subgroups of finite exponent are direct summands.
A more complete discussion of pure subgroups, their relation to infinite abelian group theory, and a survey of their literature is given in Irving Kaplansky's little red book.
[4] Since in a finitely generated abelian group the torsion subgroup is a direct summand, one might ask if the torsion subgroup is always a direct summand of an abelian group.
Under certain mild conditions, pure subgroups are direct summands.
So, one can still recover the desired result under those conditions, as in Kulikoff's paper.
Pure subgroups can be used as an intermediate property between a result on direct summands with finiteness conditions and a full result on direct summands with less restrictive finiteness conditions.
Pure subgroups were generalized in several ways in the theory of abelian groups and modules.
Pure submodules were defined in a variety of ways, but eventually settled on the modern definition in terms of tensor products or systems of equations; earlier definitions were usually more direct generalizations such as the single equation used above for n'th roots.