Definition:Zero Vector

Definition
Let $\struct {R, +_R, \times_R}$ be a ring.

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

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

The identity of $\struct {G, +_G}$ is usually denoted $\mathbf 0$, 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 $e$, or $0_V$ or $0_G$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.