Recall that the ring of Gaussian integers is a Euclidean Domain with respect to the norm
\[N(a+bi)=a^2+b^2\]
for $a+bi\in \Z[i]$.
In particular, $\Z[i]$ is a Principal Ideal Domain (PID).

Since $I$ is a nonzero ideal of the PID $\Z[i]$, there exists a nonzero element $\alpha\in \Z[i]$ such that $I=(\alpha)$.
Let $a+bi+I$ be an arbitrary element in the quotient $\Z[i]/I$.
The Division Algorithm yields that
\[a+bi=q\alpha+r,\]
for some $q, r\in \Z[i]$ and $N(r) < N(\alpha)$.

Since $a+bi-r=q\alpha \in I$, we have
\[a+bi+I=r+I.\]
It follows that every element of $\Z[i]/I$ is represented by an element $r$ whose norm is less than $N(\alpha)$.

There are only finitely many elements in $\Z[i]$ whose norm is less than $N(\alpha)$.

(There are only finitely many integers $a, b$ satisfying $a^2+b^2 < N(\alpha)$.)

The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain
Prove that the ring of integers
\[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\]
of the field $\Q(\sqrt{2})$ is a Euclidean Domain.
Proof.
First of all, it is clear that $\Z[\sqrt{2}]$ is an integral domain since it is contained in $\R$.
We use the […]

Ring of Gaussian Integers and Determine its Unit Elements
Denote by $i$ the square root of $-1$.
Let
\[R=\Z[i]=\{a+ib \mid a, b \in \Z \}\]
be the ring of Gaussian integers.
We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to
\[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\]
Here $\bar{\alpha}$ is the complex conjugate of […]

Every Prime Ideal in a PID is Maximal / A Quotient of a PID by a Prime Ideal is a PID
(a) Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal.
(b) Prove that a quotient ring of a PID by a prime ideal is a PID.
Proof.
(a) Prove that every PID is a maximal ideal.
Let $R$ be a Principal Ideal Domain (PID) and let $P$ […]

In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal
Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$.
Show that $P$ is a maximal ideal in $R$.
Definition
A commutative ring $R$ is a principal ideal domain (PID) if $R$ is a domain and any ideal $I$ is generated by a single element […]

Characteristic of an Integral Domain is 0 or a Prime Number
Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.
Definition of the characteristic of a ring.
The characteristic of a commutative ring $R$ with $1$ is defined as […]

Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals
Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.
Solution.
We give several examples. The key facts are:
An ideal $I$ of $R$ is prime if and only if $R/I$ is an integral domain.
An ideal $I$ of […]

A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring
Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.
Let $I$ be the subset of $R$ defined by
\[I:=\{ f(x) \in R \mid f(1)=0\}.\]
Then prove that $I$ is an ideal of the ring $R$.
Moreover, show that $I$ is maximal and determine […]

Three Equivalent Conditions for a Ring to be a Field
Let $R$ be a ring with $1$. Prove that the following three statements are equivalent.
The ring $R$ is a field.
The only ideals of $R$ are $(0)$ and $R$.
Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.
Proof. […]