# Definition:Ring (Abstract Algebra)/Ring Axioms

< Definition:Ring (Abstract Algebra)(Redirected from Definition:Ring Axioms)

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## Definition

A ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((A0)\) | $:$ | Closure under addition | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a * b \in R \) | ||||

\((A1)\) | $:$ | Associativity of addition | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \paren {a * b} * c = a * \paren {b * c} \) | ||||

\((A2)\) | $:$ | Commutativity of addition | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a * b = b * a \) | ||||

\((A3)\) | $:$ | Identity element for addition: the zero | \(\displaystyle \exists 0_R \in R: \forall a \in R:\) | \(\displaystyle a * 0_R = a = 0_R * a \) | ||||

\((A4)\) | $:$ | Inverse elements for addition: negative elements | \(\displaystyle \forall a \in R: \exists a' \in R:\) | \(\displaystyle a * a' = 0_R = a' * a \) | ||||

\((M0)\) | $:$ | Closure under product | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a \circ b \in R \) | ||||

\((M1)\) | $:$ | Associativity of product | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | ||||

\((D)\) | $:$ | Product is distributive over addition | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \) | ||||

\(\displaystyle \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \) |

These criteria are called the **ring axioms**.

## Also presented as

These can also be presented as:

\((A)\) | $:$ | $\struct {R, *}$ is an abelian group | ||||||

\((M0)\) | $:$ | $\struct {R, \circ}$ is closed | ||||||

\((M1)\) | $:$ | $\circ$ is associative on $R$ | ||||||

\((D)\) | $:$ | $\circ$ distributes over $*$ |

## Also see

## Sources

- 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 1.1$: The definition of a ring: Definitions $1.1 \ \text{(c)}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 54$. The definition of a ring and its elementary consequences