Can we define a $\cdot$ operation such as every abelian group (V,+) and every commutative field (K,+,$\cdot$) form a vector space

Question

As the title says, I would like to know if we can define an operation $\cdot$ such as every abelian group and commutative field form a vector space.

Since $\cdot$ is a function, $\cdot : VK \rightarrow X$, I think one can define $\cdot$ in the following manner: $$\cdot(\alpha,v) = v, if \ \alpha = 1_K$$ $$\cdot(\alpha,v) = 0, if \ \alpha \in K - \{ 1_K\}$$

By this we would define the "trivial" operation $\cdot$ (ab = 0 ,$\forall$ a,b) and we would also be able to satisfy the $1_K\cdot v = v$ axiom of vector spaces, I believe.

Unfortunately, this is all I've been able to come up with (and I'm not even sure if it this definition stands true). Any help is much appreciated!

Answer 1

Your definition doesn't work. You haven't verified the vector space axioms. According to your definition, we would have to have $2v=0$ and $3v=0$ (assuming $2,3 \ne 1$ in the field $K$), and then subtracting the 2 equations, the distributive law gives $1v=0$. But your definition says $1v=v$ for all $v$.