# Definition:Vector Addition

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## Definition

### Vector Addition on Module

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

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

Let $M := \struct {G, +_G, \circ}_R$ be the corresponding **module over $R$** (either a left module or a right module).

The group operation $+_G$ on $M$ is known as **vector addition** on $M$.

### Vector Addition on Vector Space

Let $\struct {F, +_F, \times_F}$ be a field.

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

Let $V := \struct {G, +_G, \circ}_R$ be the corresponding **vector space over $F$**.

The group operation $+_G$ on $V$ is known as **vector addition** on $V$.

## Also known as

**Vector addition** is also known as **composition**.

## Also see

- Results about
**vector addition**can be found**here**.