TORSION-FREE ABELIAN-GROUPS TORSION OVER THEIR ENDOMORPHISM-RINGS

We use a variation on a construction due to Corner 1965 to construct (Abelian) grops 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 I A = A for each maximal ideal I in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable R-module A such that (1) R = End (A), (2) T0 x(R) A not-equal 0 for each nonzero finitely generated (respectively finitely presented) R-module T0, but (3) T x(R) A = 0 for some nonzero (respectively nonzero finitely generated) R-module T.