A Counter Examples in Linear Algebra (Vector Space)

Question

I have been studying linear algebra for a few years now and in fact, also teaching it. What makes learning (and teaching) mathematics more interesting is to find examples and/or counterexamples of what we learn. As a process, I am trying to find counterexamples of sets along with two operations $+$ and $\cdot$, which we will call for the time being "addition" and "scalar multiplication" which do not form a vector space because they fail to satisfy exactly one of the axioms of a vector space.

If I am to list out the axioms and hence the definition, it proceeds as follows:-

A set $V$ along with two operations $+: V \times V \rightarrow V$ and $\cdot: \mathbb{F} \times V \rightarrow V$, where $\mathbb{F}$ is a field, is called a "vector space" over the field $\mathbb{F}$ if:

1. $\forall x, y, z \in V$ we have $\left( x + y \right) + z = x + \left( y + z \right)$
2. $\forall x, y, \in V$ we have $x + y = y + x$
3. $\exists 0 \in V$ such that $\forall x \in V$, we have $x + 0 = x$
4. $\forall x \in V$, $\exists y \in V$ such that $x + y = 0$
5. $\forall x, y \in V$ and $\forall \alpha \in \mathbb{F}$, we have $\alpha \cdot \left( x + y \right) = \alpha \cdot x + \alpha \cdot y$
6. $\forall x \in V$ and $\forall \alpha, \beta \in \mathbb{F}$, we have $\left( \alpha + \beta \right) \cdot x = \alpha \cdot x + \beta \cdot y$
7. $\forall x \in V$ and $\forall \alpha, \beta \in \mathbb{F}$, we have $\alpha \cdot \left( \beta \cdot x \right) = \left( \alpha \beta \right) \cdot x$, where $\alpha \beta$ denotes the multiplication of $\alpha$ with $\beta$ in the field $\mathbb{F}$
8. $\forall x \in V$, we have $1 \cdot v = v$, where $1 \in \mathbb{F}$ is the unity.

It is not so difficult (if not easy) to find counterexamples of sets, fields and operations which satisfy all but one property from $1$ through $7$. However, I have not yet been able to find an example which satisfy all properties except $8$ and hence fails to be a vector space. I would like some help in constructing such a counter example.

Answer 1

Take any nontrivial abelian group $V$ and any field $\mathbb{F}$, and define $c\cdot v = 0_V$ for all $c \in \mathbb{F}$ and $v\in V$.