Ring

definitionring-theory

Rings generalize fields.

Definition

We define $(R,+,\cdot )$ as a ring if $R$ is a set and,

1. $(R,+)$ is a commutative group (identity element $0=0_R$).
2. $\cdot$ is an operation on $R$:
   • $\cdot$ is associative.
   • Distributivity holds
     • $a\cdot (b+c)=a\cdot b+a\cdot c$.
     • $(a+b)\cdot c=a\cdot c+b\cdot c$.

Furthermore, if,

• $\cdot$ is commutative, we say $R$ is a commutative ring.
• $\cdot$ has identity element $1_R=1$, say $R$ is a unital ring.
• $R$ is unital and $\forall x\ne 0,\exists x^{-1}\in R$, $x\cdot x^{-1}=x^{-1}\cdot x=1_R$, say $R$ is a division ring.
• $R$ is a commutative division ring, say $R$ is a Field.

Examples

1. $Q=\{a+bi+cj+dk\mid a,b,c,d\in \mathbb{R}\}$ is the quaternion division ring.