Definition:Vector Space/Definition 2

Definition
Let $\struct {K, +_K, \times_K}$ be a field whose unity is $1_K$.

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {\map {\mathrm {End} } G, +, \circ}$ be the endomorphism ring of $\struct {G, +_G}$ such that $I_G$ is the identity mapping.

Let $\cdot: \struct {K, +_K, \times_K} \to \struct {\map {\mathrm {End} } G, +, \circ}$ be a ring homomorphism from $K$ to $\map {\mathrm {End} } G$ which maps $1_K$ to $I_G$.

Then $\struct {G, +_G, \cdot, K}$ is a vector space over $K$ or a $K$-vector space.

Also see

 * Equivalence of Definitions of Vector Space over Field