TITLE: Abelian groups injective over their endomorphism
QUESTION [6 upvotes]: Let $M$ be an abelian group and let $R = \mbox{End}_\Bbb{Z}(M)$. Under what conditions (on $M$), $_RM$ is injective!?

REPLY [7 votes]: It seems that such modules where classified by Fred Richman and Elbert A. Walker, in Modules over PIDs that are injective over their endomorphism rings, Ring Theory (Proc. Conf., Park City, Utah, 1971), p.363-372, Academic Press, N.-Y., 1972.
The paper is available on E. A. Walker's website.