Let $\mathbb{Z}[i,\epsilon] \simeq \mathbb{Z}[i](\epsilon)/(\epsilon^2)\simeq \mathbb{Z}[x,\epsilon]/(x^2 +1, \epsilon^2)$ be the Gaussian integers with an infinitesimal number $\epsilon^2 = 0$. What are the primes ideals in this ring?
- The primes in $\mathbb{Z}$ split over $\mathbb{Z}[i]$. For example, $37 = (6+i)(6-i)$ and that $\mathfrak{p} = (6 \pm i)$ is a prime ideal.
Are there any "non-trivial" prime ideals in the ring $A = \mathbb{Z}[i](\epsilon)/(\epsilon^2)$ - which is a deformation of the $\mathbb{Z}[i]$.
Do we get any extra primes?
- an ideal $\mathfrak{p}$ is prime if $\mathfrak{p} \neq 1$ and if $xy \in \mathfrak{p}$ $\to$ $x \in \mathfrak{p}$ or $y \in \mathfrak{p}$.
- $\mathfrak{p}$ is prime $\leftrightarrow$ $A/\mathfrak{p}$ is an integral domain (i.e. no zero-divisors).
We could write $A$ as a kind of $\mathbb{Z}$-module. Any element could be written $x = a_0 + a_1 i + a_2 \epsilon + a_3 i \epsilon$. Or possibly as $4 \times 4$ integer matrices. Hopefully I have written the nilpotent elements correctly.
$$ a_0 + a_1 i + a_2 \epsilon + a_3 i \epsilon \mapsto \left[ \begin{array}{cr|cr} a_0 & -a_1 & a_2 & -a_3 \\ a_1 & a_0 & a_3 & a_2 \\ \hline 0 & 0 & a_0 & -a_1 \\ 0 & 0 & a_1 & a_0 \end{array} \right] $$ I am asking about how factorizations work in this domain.
Let $R$ be your ring.
If $p$ is prime, then $R/p$ is integral so $\epsilon^2=0\implies \epsilon =0$ in $R/p$, so $\epsilon\in p$. So $p$ is of the form $\pi^{-1}(q)$ for some prime $q$ of $\mathbb{Z}[i]$, where $\pi:R\to \mathbb{Z}[i]$, $\epsilon\mapsto 0$.
Therefore $p= (a,\epsilon)$ where $a$ is a prime number in $\mathbb{Z}[i]$ or $p=(\epsilon)$.