Two-sided Ideal

Summary

Ideals are a generalisation of numbers. They have been introduced to be a replacement for unique prime factor decomposition in Nicht-faktoriellen Ringen. For example, in $\mathbb{Z}[\sqrt{5}]$, the number $6$ has two decomposition into Irreduzibles Element: $6=2\cdot 3=(1+\sqrt{-5})\cdot (1-\sqrt{-5})$ However, if you combine two numbers into an ideal $(2,1+\sqrt{-5})$, then the decomposition of the ideal $(6)$ is unique. I think…

They are the analogue of the Normalteiler in group theory, as their definition is similar and they allow Faktorstrukturen.

Definition

Let $R$ be a possibly Noncommutative Ring. An Ideal in $R$ is a subset $I\subseteq R$ fulflilling

1. $0_R\in I$
2. Für alle $a,b\in I$ und $r\in R$ gilt $a+b\in I$ und $ra,ar\in I$

If $R$ is a Commutative Ring, then the Left ideal, right ideal and two-sided ideal is the same.

Properties

Genereating of new ideals

• The possibly infinite intersection of ideals is an ideal.
• The Sum of two ideals $I+J=\{a+b:a\in I,b\in J\}$ is an ideal
• The Produktideal is an ideal

Characterisation of Subsets

There are some possible characterisations of subideals

1. $(a)\subset (b)$ holds i.f.f. $b|a$
2. If $d\in R$ where $(d)=(a,b)=(a)+(b)$, then is $d$ the gcd of $a$ and $b$
3. If $e\in R$ where $(e)=(a)\cap (b)$, then $e$ is a gcd of $a$ and $b$

In a Principal Ideal Domain the inversions of ii und iii hold as well.[^1]

The correspondence theorem states that image and preimage of ideals under Ring homomorphism are ideals as well. And their subset-ordering is preserved.

Correspondence theorem

Let $R$ be a Ring, $I$ an ideal and $\pi :R\to R/I$ the canonical epimorphism. Let $\tilde{\mathcal{I}}$ be the set of ideals of $R/I$ and ${\mathcal{I}}_I$ the set of ideals $J$ bigger then $I$, i.e. $J\supseteq I$. The following holds:

1. The mappings $\varphi :{\mathcal{I}}_I\to \tilde{\mathcal{I}},J\mapsto \pi (J)$ and $\psi :\tilde{\mathcal{I}}\to {\mathcal{I}}_I,\tilde{J}\mapsto {\pi }^{-1}(J)$ are bijective
2. For all ideals $J,K\in {\mathcal{I}}_I$ we have $J\subseteq K⟺\pi (J)\subseteq \pi (K)$

Examples

Hauptideal

Siehe Two-sided Principal ideal

Trivial Ideal

In every Commutative Ring $R$ the Zero ideal $(0_R)=\{0_R\}$ is the smalles and the unit ideal $(1_R)=R$ the biggest Ideal.