Definition:Vector Space

Let $$\left({K, +_K, \times_K}\right)$$ be a division ring.

Let $$\left({G, +_G}\right)$$ be an abelian group.

Let $$\left({G, +_G: \circ}\right)_K$$ be a unitary $K$-module.

Then $$\left({G, +_G: \circ}\right)_K$$ is a vector space over $$K$$ or a $$K$$-vector space.

That is, a vector space is a unitary module whose scalar ring is a division ring.

Vector
The elements of $$\left({G, +_G}\right)$$ are called vectors.

Zero Vector
The identity of $$\left({G, +_G}\right)$$ is usually denoted $$0$$, or some variant of this, and called the zero vector.

Note that on occasion it is advantageous to denote the zero vector differently, for example by $$e$$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.

Comment
As a vector space is also a unitary module, all the results which apply to modules, and to unitary modules, also apply to vector spaces.