# Definition:Commutative and Unitary Ring

## Definition

A **commutative and unitary ring** $\struct {R, +, \circ}$ is a ring with unity which is also commutative.

That is, it is a ring such that the ring product $\struct {R, \circ}$ is commutative and has an identity element.

That is, such that the multiplicative semigroup $\struct {R, \circ}$ is a commutative monoid.

The identity element is usually denoted by $1_R$ or $1$ and called a unity.

### Commutative and Unitary Ring Axioms

A commutative and unitary ring is an algebraic structure $\left({R, *, \circ}\right)$, 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 | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right) \) | ||||

\((A2)\) | $:$ | 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 | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right) \) | ||||

\((M2)\) | $:$ | Commutativity of product | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a \circ b = b \circ a \) | ||||

\((M3)\) | $:$ | Identity element for product: the unity | \(\displaystyle \exists 1_R \in R: \forall a \in R:\) | \(\displaystyle a \circ 1_R = a = 1_R \circ a \) | ||||

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

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

These criteria are called the **commutative and unitary ring axioms**.

## Also known as

Other nomenclature includes:

**Commutative and unital ring****Commutative ring with unity****Commutative ring with identity**

## Also see

- Results about
**commutative and unitary rings**can be found here.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 18$. Definition of a Ring - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields