Commutative ring/Ideal/Introduction/Section

Definition

A subset ${}{\mathfrak {a}}$ of a commutative ring ${}R$ is called an ideal, if the following conditions are fulfilled:

1. ${}0\in {\mathfrak {a}}$.
2. For all ${}a,b\in {\mathfrak {a}}$, we have ${}a+b\in {\mathfrak {a}}$.
3. For all ${}a\in {\mathfrak {a}}$ and ${}r\in R$, we have ${}ra\in {\mathfrak {a}}$.

The property ${}0\in {\mathfrak {a}}$ can be replaced by the condition that ${}{\mathfrak {a}}$ is not empty. An ideal is a subgroup of the additive group of ${}R$, which, moreover, is also closed under scalar multiplication.

Definition

For a family of element ${}a_{1},a_{2},\ldots ,a_{n}\in R$ in a commutative ring ${}R$, we denote by ${}(a_{1},a_{2},\ldots ,a_{n})$ the ideal generated by these elements. It consists of all linear combinations

$r_{1}a_{1}+r_{2}a_{2}+\cdots +r_{n}a_{n},$

where

${}r_{1},r_{2},\ldots ,r_{n}\in R$.

Definition

An ideal ${}{\mathfrak {a}}$ in a commutative ring ${}R$ of the form

${}{\mathfrak {a}}=(a)=Ra=\{ra:\,r\in R\}\,$

is called a principal ideal.

The zero element forms, in every ring, the so-called zero ideal. We write this simply as ${}0=(0)=\{0\}$. write. The ${}1$ and, moreover, every unit, generates as an ideal the total ring.

Definition

The unit ideal in a commutative ring

${}R$ is the ring itself.

In a field, there exist exactly two ideals.

Lemma

Let ${}R$ be a commutative ring. Then the following statements are equivalent.

1. ${}R$ is a field.
2. There exist exactly two ideals in ${}R$.

Proof

If ${}R$ is a field, then there exists the zero ideal and the unit ideal, and these are different ideals. Let ${}I$ be an ideal in ${}R$ different from ${}0$. Then ${}I$ contains some element ${}x\neq 0$, which is a unit. Therefore, ${}1=xx^{-1}\in I$ and thus ${}I=R$.

Suppose now that ${}R$ is a commutative ring with exactly two ideals. Then ${}R$ is not the zero ring. Let now ${}x$ be an element in ${}R$ different from ${}0$. The principal ideal generated by ${}x$, that is ${}Rx$, is ${}\neq 0$, and therefore it must be the other ideal, which is the unit ideal. In particular, this means ${}1\in Rx$. Hence, ${}1=xr$ for some ${}r\in R$, so that ${}x$ is a unit.

$\Box$