Give an example of an ideal of $\mathbb{Z}[i]$ known as the ring of Gaussian integers.
Solution
We know that ideals are difficult to be found, but we know how to construct one. Let $\mathcal{R}$ be a ring and let $r$ be an element of it, then $r \mathcal{R}$ is an ideal of the ring $\mathcal{R}$. Hence an ideal of $\mathbb{Z}[i]$ is:
$$(1+i) \mathbb{Z}[i]$$
Some comments: $\mathbb{Z}[i]$ is a domain of prime ideals and as if that wasn't enough it is also a domain that has unique factorization.For example the Gauss integer $1+i$ is analyzed uniquely, while the numbers $3-i$ and $13$ are not. Also there are infinite ideals in this ring.
Solution
We know that ideals are difficult to be found, but we know how to construct one. Let $\mathcal{R}$ be a ring and let $r$ be an element of it, then $r \mathcal{R}$ is an ideal of the ring $\mathcal{R}$. Hence an ideal of $\mathbb{Z}[i]$ is:
$$(1+i) \mathbb{Z}[i]$$
Some comments: $\mathbb{Z}[i]$ is a domain of prime ideals and as if that wasn't enough it is also a domain that has unique factorization.For example the Gauss integer $1+i$ is analyzed uniquely, while the numbers $3-i$ and $13$ are not. Also there are infinite ideals in this ring.
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