Commutative Ring

Description

A commutative ring (sometimes just ring) is a mathematical structure which allows addition and multiplication of its elements. Differently than a Noncommutative Ring the multiplication is commutative. It has an important role in number theory.

Definition

A commutative ring is a tripel $(R,+,\cdot )$ composed of a set $R$ and two operations $+:R\times R\to R$ and $\cdot :R\times R\to R$, called addition and multiplication. They fulfill the following porperties:

1. $(R,+)$ is an abelian group
2. $(R,\cdot )$ is a commutative Monoid (i.e. an inverse is not required)
3. The distributive law holds $a(b+c)=ab+ac$

Definition (Zero, One)

• The neutral element of addition is called zero, denoted $0$
• The neutral element of multiplication is called one, denoted $1$

Notation

We use the additive notation for addition. This means, if $R$ is a ring, $r\in R$ und $n\in \mathbb{Z}$, then $n\cdot r$ is the $n$-times addition. e.g.: $3r=r+r+r$

Classification

Rings are generally classified by specialisations. The more general the less unique prime factorisation becomes.

Ring type	What part of Prime factorisation is fixed?	Example, which is not an example of the next class
Noncommutative Ring	Almost no structure	Matrixgruppe
Commutative Ring	Factors are commutative	$\mathbb{Z}/4\mathbb{Z}$, since $2$ is a zero divisor
Integral domain	$0$ has no factorisation. Every ideal has a unique factorisation into prime ideals. (possibly)	$\mathbb{Z}[\sqrt{-5}]$, since $6$ has no unique factorisation into irreducible elements
Unique factorization domain	Irreduzibles Element are equal to the Prime element Therefore factorisation into irreducible elements are prime factorisations.	$\mathbb{Z}[x]$, since $(2,x)$ is no Principal ideal
Principal Ideal Domain	Every Ideal is a Principal Ideal Domain, meaning we can talk about factorisations into numbers instead of Prime ideal.	$\mathbb{Z}[\frac{1}{2}(1+\sqrt{-19})]$
Euclidian ring	The Euklidischer Algorithmus terminates and returns a prime number	$\mathbb{Z}[i]$
$\mathbb{Z}$	Is the most fundamental ring as there exists a homeomorphism from $\mathbb{Z}$ to every other ring.

Properties

Base ring

$\mathbb{Z}$ seems to be the most fundamental ring as there is a Ring homomorphism $\mathbb{Z}\to R$ into every other ring $R$. This homeomorphism is obtained by mapping $0\mapsto 0_R$ and $1\mapsto 1_R$

Examples

Zero ring

The triple $(\{0\},+,\cdot )$. This should be the only ring where the one is equal to the zero.