Faithful Abelian Groups of Infinite Rank
Copyright policy )Faithful Abelian Groups of Infinite Rank
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.
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Direct sums, direct products, etc. for abelian groups, Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups, torsion-free abelian group, direct sum, epimorphism, Torsion-free groups, infinite rank, Endomorphism rings; matrix rings
Direct sums, direct products, etc. for abelian groups, Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups, torsion-free abelian group, direct sum, epimorphism, Torsion-free groups, infinite rank, Endomorphism rings; matrix rings
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