Ring

algebra

Definition

Ring

A ring $(R,+,\cdot )$ is an algebraic structure consisting of a set $R$ and two binary operations satisfying:

1. $(R,+)$ is an abelian group.
2. $(R,\cdot )$ is a semigroup (associative).
3. Multiplication is distributive over addition.

$$a\cdot (b+c)=a\cdot b+a\cdot c\text{and}(a+b)\cdot c=a\cdot c+b\cdot c$$

Note: This definition does not require a multiplicative identity. Such structures are sometimes called rngs. If a multiplicative identity exists, it is a unital ring (see Ring with 1).

Variants

There are several extensions of the definition of a ring, such as the following:

Commutative Ring

Definition

Commutative Ring

A commutative ring is a ring $(R,+,\cdot )$ in which the multiplication operation is commutative.

$$\forall a,b\in R:a\cdot b=b\cdot a$$

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Ring with 1

Definition

Ring with 1

A ring with 1 (or unital ring) is a ring $(R,+,\cdot )$ that contains a multiplicative identity element, denoted as $1$.

$$\exists 1\in R:\forall a\in R,1\cdot a=a\cdot 1=a$$

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