A non trivial system verifying some statements 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  :

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

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

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

$\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?
Here's a recipe for a more general class of counterexamples. Start with a vector space $V$ over $\Bbb{F}$ (with operations $+, \cdot$). Fix a linear projection operator $\phi$ on $V$ (i.e. $\phi^2 = \phi$). Define a new scalar multiplication
$$\alpha \odot v = \alpha \cdot \phi(v).$$
Then $\odot$ satisfies all the axioms except possibly the final axiom.

Note that this counterexample covers both of the counterexamples presented, plus others.
The first four axioms can be summarized to a single axiom:


  $(V,+)$ is an abelian group


The next three can be summarized as:


  $\Bbb F$, viewed as non-unitary ring, acts on $(V,+)$.


With this, let $U=\{\,1\cdot v\mid v\in V\,\}$.
Then $1\cdot u=u$ for all $u\in U$ and in fact $U=\{\,v\in V\mid 1\cdot v=v\,\}$. One wuickly check that $U$ is in fact a subgroup of $V$ and closed under scalar multiplication, hence is a vector  space.

Now let $W=\{\,v-1\cdot v\mid v\in V\,\}$. Then for $w\in W$,
$1\cdot w=1\cdot v-1\cdot 1\cdot v=0 $ and in fact $W=\{\,v\in V\mid 1\cdot w=0\,\}$.
One quickly checks that $W$ is in fact a subgroup of $V$. 

Finally one sees that $V=U\oplus W$ as abelian groups. Thus the most general counterexample is of the form $V=U\oplus W$ where $U$ is a vector space and $W$ an abelian group (non.trivial, to make it actually a counterexample) and
$\alpha\cdot (u,w):=(\alpha u,0)$.