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Let A be an abelian group, let A = End (A), and assume that A is a flat left A-module. Then

We use a variation on a construction due to Corner 1965 to construct (Abelian) groups A that are torsion as modules over the ring End {A) of group endomorphisms of A. Some applications include the failure of the Baer-Kaplansky Theorem for Z [X]. There is a countable reduced torsion-free group A such that IA = A for each maximal ideal / in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable it-module A such that (1) R = End(A), (2) T o ®R A ^ 0 for each nonzero finitely generated (respectively finitely presented) fl-module To, but (3) T ®H A = 0 for some nonzero (respectively nonzero finitely generated) .R-module T.