Motivation

Recall that in chapter 1, §1.4, we studied the following question:

Question: Let $a$ and $b$ be any two non-zero integers. Can the set of all possible sums of multiples of $a$ and $b$ be described in a ``simple'' way? If so, how?

Before we recall how we answered this question, let us introduce some terminology. The `` the set of all possible sums of multiples'' of a given set of elements occurs so often in abstract algebra that it has a special name.

Definition 2.3.1   Let $R$ be a ring and let $a_1,...,a_k$ be elements of $R$. The set $I$ of all possible sums of multiples of the $a_i$ is called the ideal of $R$ generated by $a_1$, ..., $a_k$. In other words,

$$I=\{r_1a_1+...r_ka_k\ \vert\ r_i\in R\}.$$

When there is no danger of confusion, it is often denoted $I=(a_1,...,a_k)$.

In this terminology, the question above becomes: What is $I=(a,b)\subset \mathbb{Z}$?

We answered the question by proving that any subset of the integers which was closed under addition and (integer) multiplication must be of the form $c\mathbb{Z}$, for some integer $c\geq 1$ (Lemma 1.4.2). (In fact, $c=gcd(a,b)$.) In other words, we showed $I=(a,b)=(c)$, where $c=gcd(a,b)$. This may be reworded as saying, the ideal in $\mathbb{Z}$ generated by the $2$ elements $a,b$ is an ideal generated by only $1$ element $c=gcd(a,b)$. Ideals generated by only one elements also have a special name.

Definition 2.3.2   Let $R$ be a ring and let $a$ be an element of $R$. The set $I$ of all possible multiples of $a$ is called the principal ideal of $R$ generated by $a$. In other words,

$$I=\{ra\ \vert\ r_i\in R\}.$$

When there is no danger of confusion, it is often denoted $I=(a)$.

Using this terminology, we showed in chapter 1 that the ideal of $\mathbb{Z}$ generated by $a,b$ is a principal ideal. One can use mathematical induction to show that any ideal in $\mathbb{Z}$ of the for $I=(a_1,...,a_k)$ is a principal ideal. Then every ideal of a ring $R$ is a principal ideal, then $R$ is called a principal ideal domain. It is possible to show that the ring $\mathbb{Z}$ is a principal ideal domain.