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 $1.1 \ \text{(c)}$