# Axiom: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:

\((\text A 0)\) | $:$ | Closure under addition | \(\ds \forall a, b \in R:\) | \(\ds a * b \in R \) | ||||

\((\text A 1)\) | $:$ | Associativity of addition | \(\ds \forall a, b, c \in R:\) | \(\ds \paren {a * b} * c = a * \paren {b * c} \) | ||||

\((\text A 2)\) | $:$ | Commutativity of addition | \(\ds \forall a, b \in R:\) | \(\ds a * b = b * a \) | ||||

\((\text A 3)\) | $:$ | Identity element for addition: the zero | \(\ds \exists 0_R \in R: \forall a \in R:\) | \(\ds a * 0_R = a = 0_R * a \) | ||||

\((\text A 4)\) | $:$ | Inverse elements for addition: negative elements | \(\ds \forall a \in R: \exists a' \in R:\) | \(\ds a * a' = 0_R = a' * a \) | ||||

\((\text M 0)\) | $:$ | Closure under product | \(\ds \forall a, b \in R:\) | \(\ds a \circ b \in R \) | ||||

\((\text M 1)\) | $:$ | Associativity of product | \(\ds \forall a, b, c \in R:\) | \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | ||||

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

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

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

## Also presented as

These can also be presented as:

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

\((\text M 0)\) | $:$ | $\struct {R, \circ}$ is closed | ||||||

\((\text M 1)\) | $:$ | $\circ$ is associative on $R$ | ||||||

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

## Also defined as

For a **ring with unity**, the following axiom also holds:

\((\text M 2)\) | $:$ | Identity element for $\circ$: the unity | \(\ds \exists 1_R \in R: \forall a \in R:\) | \(\ds a \circ 1_R = a = 1_R \circ a \) |

For a **commutative ring**, the following axiom also holds:

\((\text C)\) | $:$ | Commutativity of Ring Product | \(\ds \forall a, b \in R:\) | \(\ds a \circ b = b \circ a \) |

## Also see

## Sources

- 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 $1.1 \ \text{(c)}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 54$. The definition of a ring and its elementary consequences