# Example of a field $F$ and a structure $(V,+,s)$ satisfying all but one of the vector space axioms.

I have to find an example (if there exists one) of a field $$F$$ and a structure $$(V,+,s)$$ (where $$s$$ stands for scalar multiplication) satisfying all but one of the axioms of a vector space, namely the distributivity of vector addition: For all $$c ∈ F$$ and $$v,w ∈ V$$ , $$c(v + w) = cv + cw$$

I believe such example exists, but I'm not sure where to start. An exercise I did before asked to show how the axiom follows from the other axioms if $$F=\mathbb{Q}$$. So I think perhaps using a different field like $$\mathbb{R}$$ (or anything similar) could be of use.

Could anyone give me some advice or
hint to solve it?

Thanks!

• "namely the distributivity of vector addition." - To be clear, is this the axiom you mean? ($a \in F; \vec v, \vec w \in V$; scalar multiplication denoted by $\cdot$): $$a \cdot (\vec v + \vec w) = a \cdot \vec v + a \cdot \vec w$$ Aug 6, 2020 at 2:32
• I think so. I see none other.
– user732848
Aug 6, 2020 at 2:33
• Yes, that's the axiom Aug 6, 2020 at 2:38
• Maybe math.stackexchange.com/questions/1412899/… will do. Aug 6, 2020 at 4:15
• – lhf
Aug 6, 2020 at 10:23