Is this set neccesarily to be a vector space? Suppose $F$ be a field and $S$ be a non empty set such that
1) $a+b \in S $
2) $ \alpha a \in S$
for all $a,b \in S$ and $ \alpha \in F.$
Is always $S$ to be a Vector space?

No, for we could take the multiplication map $\cdot: F \times S$ to be zero map---the resulting structure fails to be a vector space only because it does not have a multiplicative identity, i.e. an element $1 \in F$ such that $1 \cdot s = s$ for all $s \in S$.
Ferra's nice example shows that preventing this degenerate solution by demanding existence of a multiplicative identity still isn't enough to guarantee that the $S$ is a vector space over $F$.
