Example

Ideals and quotient rings of $\mathbf{Z}$

An ideal of $\mathbf{Z}$ is an additive subgroup of $\mathbf{Z}$ and hence of the form $n.\mathbf{Z}$ with $n \geqslant 0$; conversely, for every integer $n \geqslant 0$, the set $n.\mathbf{Z}$ is an ideal, the principal ideal $(n)$. Thus every ideal of $\mathbf{Z}$ is principal and is represented uniquely in the form $n\mathbf{Z}$ with $n \geqslant 0$. The ideal $(1)$ is equal to $\mathbf{Z}$, the ideal $(0)$ consists of 0 and the ideals distinct from $\mathbf{Z}$ and $\{0\}$ are therefore of the form $n\mathbf{Z}$ with $n > 1$. If $m \geqslant 1$ and $n \geqslant 1$, $m\mathbf{Z} \supset n\mathbf{Z}$ if and only if $n \in m.\mathbf{Z}$, that is $m$ divides $n$. Therefore, for the ideal $n\mathbf{Z}$ to be maximal, it is necessary and sufficient that there exist no integer $m > 1$ distinct from $n$ and dividing $n$; in other words, the maximal ideals of $\mathbf{Z}$ are the ideals of the form $p\mathbf{Z}$ where $p$ is a prime number (§ 4, no. 10, Definition 16).

Let $m$ and $n$ be two integers $\geqslant 1$. The ideal $m\mathbf{Z} + n\mathbf{Z}$ is principal, whence there is an integer $d \geqslant 1$ characterized by $d\mathbf{Z} = m\mathbf{Z} + n\mathbf{Z}$; for every integer r \geqslant 1, the relation "r divides d" is equivalent to $r\mathbf{Z} \supset d\mathbf{Z}$ and hence to "$r\mathbf{Z} \supset m\mathbf{Z}$ and $r\mathbf{Z} \supset n\mathbf{Z}$", that is to "r divides m and n". It is thus seen that the common divisors of m and n are the divisors of d and that d is the greatest of the divisors $\geqslant 1$ common to m and n; d is called the greatest common divisor (abbreviated to g.c.d.) of m and n. As $d\mathbf{Z} = m\mathbf{Z} + n\mathbf{Z}$, there exist two integers x and y such that $d = mx + ny$. m and n are said to be relatively prime if their g.c.d. is equal to 1. It amounts to the same to assume that there exist integers x and y with $mx + ny = 1$.

The intersection of the ideals $m\mathbf{Z}$ and $n\mathbf{Z}$ is non-zero for it contains $mn$ and hence is of the form $r\mathbf{Z}$ with $r \geqslant 1$. Arguing as above, it is seen that the multiples of r are the common multiples of m and n and that r is the least of the integers $\geqslant 1$ which are common multiples of m and n; it is called the least common multiple (l.c.m.) of m and n.

The product of the ideals $m\mathbf{Z}$ and $n\mathbf{Z}$ is the set of $\sum_{i=1}^{r} mx_i ny_i = mn \left( \sum_{i=1}^{r} x_i y_i \right)$ for $x_1, \ldots, y_r \in \mathbf{Z}$ and hence is equal to $mn\mathbf{Z}$.

For every integer $n \geqslant 1$, the quotient ring $\mathbf{Z}/n\mathbf{Z}$ is called the ring of integers modulo n; it has n elements, which are the classes modulo n of the integers 0, 1, 2, ..., $n - 1$. For $n = 1$, we obtain the zero ring.