# Definition:Ring (Abstract Algebra)/Addition

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## Definition

The distributand $*$ of a ring $\struct {R, *, \circ}$ is referred to as **ring addition**, or just **addition**.

The conventional symbol for this operation is $+$, and thus a general ring is usually denoted $\struct {R, +, \circ}$.

### Additive Group

The group $\struct {R, +}$ is known as the **additive group of $R$**.

### Additive Inverse

Let $\struct {R, +, \circ}$ be a ring whose ring addition operation is $+$.

Let $a \in R$ be any arbitrary element of $R$.

The **additive inverse** of $a$ is its inverse under ring addition, denoted $-a$:

- $a + \paren {-a} = 0_R$

where $0_R$ is the zero of $R$.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(c)}$