Definition – Ring. – Math-reference.net

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1. Definition

Definition

Let $R$ be a set with $\times$ and $+$ both binary operations. Then, we say that the set $R$ with operations $\times$ and $+$ are a ring if the following are true,

$\langle R, + \rangle$ is an abelian group.
The set $R$ with the operation $\times$ is associative.
The operation $\times$ is distributive over the $+$ operation.
There exists an element $1_R$ such that $\forall r\in R$ , $r\times 1_R = 1_R\times r = r$ .

If $R$ with $\times$ is commutative, we refer to the ring as a commutative ring.

Types of ring

As above, if $p\times q = q \times p$ for all $p,q\in R$ , then we refer to the ring as a commutative ring.

Given a commutative ring, if $a\times b = 0$ (which is the ring’s identity element under the $+$ operation) implies that either $a=0$ or $b = 0$ , then we refer to the ring as an integral domain.

A Euclidean domain is a integral domain with a function $\sigma : R\setminus \{0\}$ such that for all $a,b\in R\setminus \{0\}$ ,

There exists $q,r \in R$ such that $a = bq + r$ , with $r=0$ or $\sigma(r) < \sigma(b)$ .
If $a|b$ , then $\sigma(a) \leq \sigma (b)$ .

The function $\sigma$ can be thought of as a form of measure on the set $R$ , as it compares with the natural numbers.

Types of element

We refer to an element $r\in R$ as a unit if $r$ has a multiplicative inverse in $R$ .

We say that an element $r\in R$ is irreducible if, for $r = a\times b$ , either $a$ or $b$ is a unit.