There are two "distributivity" laws in an $R$-module/vector space:
- For all $a\in R$, $x,y\in M$, $a(x+y) = ax+ay$;
- For all $a,b\in R$, $x\in M$, $(a+b)x = ax+bx$.
An example in which all axioms of a vector space except for (1) above holds is:
Take $V=\mathbb{C}^2$ with its usual addition; define scalar multiplication by:
$$\alpha(x,y) = \left\{\begin{array}{ll}
(\alpha x,\alpha y) &\text{if }x\neq 0;\\
(0,\overline{\alpha}y) &\text{if }x=0.
\end{array}\right.$$
With an arbitrary ring/field, any nontrivial automorphism will do instead of complex conjugation.
If you don't mind other axioms failing, you can take $V=\mathbb{R}^n$ over $\mathbb{R}$, take $\alpha(x,y) = (0,0)$ if $\alpha\neq 1$, and $1(x,y)=(x,y)$, with $V=\mathbb{R}^2$. Note, however, that this does not satisfy associativity: if $\alpha\neq 0,1$ and $\beta=\frac{1}{\alpha}$, then $\alpha(\beta (x,y)) =(0,0)$, but $(\alpha\beta)(x,y) = (x,y)$.
An example in which all axioms of a vector space except for (2) above holds is: take $V=\mathbb{R}$ with its usual addition, and define scalar multiplication by $\alpha\cdot x = \alpha^2x$.
For a longer discussion of the independence of the sundry vector space axioms, see here.
