Let $B$ be a pure subgroup of an abelian group $A$. In his book "Infinite Abelian Groups", Academic Press, 2vols., Fuchs defines $A$ to be a pure-essential extension of $B$ if there is no nonzero subgroup $H$ of $A$ such that (i) $B \cap H = 0$ and (ii) $(B+H)/H$ is pure in $A/H$. In "Infinite Abelian Group Theory", University of Chicago Lectures, Griffiths makes the same definition with condition (ii) replaced by (ii)' $B+H$ is pure in $A$. Using standard facts about purity, I can see that the two definitions are equivalent if $H$ is pure in $A$, but not otherwise. Would someone be so kind as to point out what I am missing, or, if, in fact, the definitions are not equivalent?
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