Faithful abelian groups of infinite rank

Abstract

Let B B be a subgroup of an abelian group G G such that G / B G/B is isomorphic to a direct sum of copies of an abelian group A A . For B B to be a direct summand of G G , it is necessary that G G be generated by B B and all homomorphic images of A A in G G . However, if the functor Hom ⁡ ( A , − ) \operatorname {Hom} (A, - ) preserves direct sums of copies of A A , then this condition is sufficient too if and only if M ⊗ E ( A ) A M{ \otimes _{E(A)}}A is nonzero for all nonzero right E ( A ) E(A) -modules M M . Several examples and related results are given.