Let $A/F$ be an abelian variety over a number field $F$ and let $P \in A(F)$ and $\Lambda \subset A(F)$ be a subgroup of the Mordell-Weil group. For a prime $v$ of good reduction let $r_v : A(F) \rightarrow A_v (k_v)$ be the reduction map. During my talk I will show that the condition $r_v (P) \in r_v (\Lambda)$ for almost all primes $v$ implies that $P \in \Lambda + A(F)_{tor}$ for a wide class of abelian varieties.