A non trivial system verifying some statements;

Question

For every abelian group $(V,\oplus)$ and an unitary field $\left( {{\Bbb F},{\text{ }} + ,{\text{ }}{\text{.}}} \right)$ acting on $(V,\oplus)$ by the external operation $\odot$ .

Question : Is there a non trivial system such that :

1. $\forall x, y \in V, \ \ $ $\forall \alpha \in \mathbb{F}, \ \ $ $\alpha \odot \left( x \oplus y \right) = \left(\alpha \odot x \right)\oplus\left( \alpha \odot y\right)$ .

2. $\forall x \in V, \ \ $ $\forall \alpha, \beta \in \mathbb{F}, \ \ $ $\left( \alpha + \beta \right) \odot x = \left(\alpha \odot x \right)\oplus\left( \beta \odot x\right)$ .

3. $\forall x \in V, \ \ $ $\forall \alpha, \beta \in \mathbb{F}, \ \ \alpha \odot \left( \beta \odot x \right) = \left( \alpha\cdot \beta \right) \odot x$ .

4. $\exists u \in V , \ \ u \ne {0_V} \Rightarrow {{\text{1}}_{\Bbb F}} \odot {\text{ }}u \ne u $ .

   And if yes, is there a general theorem of a such system?