Vector Space

A vector space is a mathematical object $(R:\text{Ring},V:\ast ,+:V\times V\to V,\cdot :R\times V\to V,\vec{0}:V)$ where $R$ is some Ring.

It must satisfy the following axioms:

$$\begin{array}{r}\forall \vec{x}:V.\vec{x}+\vec{0}=\vec{x}\\ \forall \vec{x},\vec{y}:V.\vec{x}+\vec{y}=\vec{y}+\vec{x}\\ \forall \vec{x},\vec{y},\vec{z}:V.(\vec{x}+\vec{y})+\vec{z}=\vec{x}+(\vec{y}+\vec{z})\\ \forall \vec{x}:V.0_R\cdot x=\vec{0}\\ \forall a,b:R,\vec{x}:V.a\cdot \vec{x}+b\cdot \vec{x}=(a+b)\cdot x\end{array}\text{\hspace{1em}(1)(2)(3)(4)(5)}$$