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We wanted to understand the structure of finite rankorsion free
abelian group. But in fact, there are only few restricted
torsion free abelian groups that resulted in good conclusion.
In this paper, we describe about hom of finite rank torsion
free abelian groups and classify them by types. In generally,
the type of group desides the part of structure of it. This
paper has two parts. The first part contains 3 lemmas, 4
theorems and 2 corollaries, the important results of it are
theorem,8 and corollary,9. Theorem,8 : Suppose that A and B are
torsion free groups of rank-1. (a) If type(A)$\le$type(B) then
Hom(A,B) is a rank-1 torsion free groups with type=[($k_p$)],
where 0$\ne a\in A$, $0\ne b\in B$, $h^A(a)=(k_p)\le h^B(b)=(
l_p), m_p=\infty$ if $l_p= \infty ,and m_p=l_p - k_p if l_p is
finite. (b) If type(A) =[(k_p)] then type(Hom(A,A))=[(m_p)] is
non-nil, where m_p= \infty if k_p=\infty and m_p=0 if k_p=
\infty. Corollary,9 : Assume A is a tosion free group of
rank-1. The fllowing are equivalent : (a) A have non-nil type ;
(b) type(A)+type(A)=ype(A); (c) A\congHom(A,A); (d) A is
isomorphic to the additive group of a subring of Q; (e) If 0\ne
a\in A then A/Za=T\oplusD, where T is a finite torsion group
and D\cong \oplus_p\in S Z(p^\infty) for some subset S of \Pi.
The second part contains 2 lemmas, 5 theorems and 2
corollaries, the important conclusion of it are theorem,15 and
corollary,17, 18.
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