It is well known that the set of homomorphisms from a fixed abelian group A to a fixed abelian group B forms an additive abelian group denoted as Hom(A, B). Homomorphism groups of abelian groups possess many remarkable properties. For example, they behave like functors in the category of abelian groups. In some important cases, one can express invariants of the group Hom(A, B) in terms of invariants of the groups A and B, e.g., if A is a torsion abelian group or if B is an algebraically compact abelian group. If A = B, the group Hom(A, B) = End(A, B) is called the endomorphism group of the group A it can be turned into a ring denoted as E(A). Studying homomorphism groups and endomorphism rings is an important problem of the theory of abelian groups. In particular, describing abelian groups such that Hom(A, B) = 0 is one of open problems in this theory. For example, the group Hom(A, B) is zero in the following case. Let an abelian group G be decomposed into a sum of its subgroups A and B, A being a fully invariant subgroup in the group G, i.e., A is mapped into itself under any endomorphism of the group G. Then, Hom(A, B)= 0. The torsion subgroup of a group, for example, is its fully invariant subgroup. In this paper, a criterion of vanishing is presented for an arbitrary homomorphism from an arbitrary abelian group to an arbitrary torsion free group.

1. Grinshpon S.Ya. Problem 2 [Problema 2]. Abelevy gruppy: Trudy Vserossijskogo simpoziuma [Abelian Groups: Transaction All-Russian Symposium]. Biysk, 2005, p. 60.

2. Grinshpon S.Ya. On the Equality to Zero of the Homomorphism Group of Abelian Groups [O ravenstve nulju gruppy gomomorfizmov abelevyh grupp]. Izvestiya VUZ. Matematika [Izvestija vysshih uchebnyh zavedenij. Matematika], 1998, vol. 42, no. 9, pp. 39-43.

3. Schultz P. Annihilator Classes of Torsion-free Abelian Groups. Lect. Notes Math, 1978, vol. 697, pp. 88-94.

4. Dimitric R. On Coslender Groups. Glasnik Matem, 1986, vol.21, no 2, pp. 327-329.

5. Krylov P.A., Podberezina Ye.I. The Group Hom(A, B) as an Artinian E(B)-or E(A)-module. Journal of Mathematical Sciences, 2008, vol. 154, issue 3, pp. 333-343.

6. Mishina A.P. On the Automorphism and the Endomorphism of Abelian Groups [Ob avtomorfizmah i jendomorfizmah abelevyh grupp]. Moscow University Bulletin. Series 1. Mathematics. Mechanics [Vestnik Moskovskogo Universiteta. Serija 1. Matematika i mehanika], 1962, no 4. pp. 39-43.

7. Grinshpon S.Ya., Yeltsova T.A. Homomorphic Images of Abelian Groups. Journal of Mathematical Sciences, 2008, vol. 154, issue 3, pp. 290-294.

8. Chekhlov A.R. On Abelian Groups Close to E-solvable Groups [Ob abelevyh gruppah, blizkih k E-razreshimym gruppam]. Fundam. Prikl. Mat. [Fundamental'naja i prikladnaja matematika], 2012, vol. 17, issue 8, pp. 183-219.

9. Kulikov L.Ya. Generalized Primary Groups. II [Obobshhenno primarnye gruppy. II]. Tr. Mosk. Mat. Obs. [Trudy moskovskogo matematicheskogo obshhestva], 1953, vol. 2, pp. 85-167.

10. Fuchs L. Infinite Abelian Groups. Pure and Applied Mathematics. 36. Academic Press., New York London, 1970, vol. I.

11. Fuchs L. Infinite Abelian Groups. Pure and Applied Mathematics. 36. Academic Press, New York London, 1973, vol. II.