How to find the Spectrum of a special matrix algebra?

Question

I have the following question:

How to find all prime ideals of the $\mathbb{Z}$-algebra $\begin{equation} \Lambda= \begin{pmatrix} \mathbb{Z} & 3\mathbb{Z} \\ \mathbb{Z} & \mathbb{Z} \end{pmatrix} \end{equation}$?

Thanks in advance for the help.

Answer 1

Observations

Let $I\lhd \Lambda$. Then $\left[\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right]\Lambda\left[\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right]$ is a subgroup of $\left[\begin{smallmatrix}0&3\Bbb Z\\0&0\end{smallmatrix}\right]$, and we know very well that the possible subgroups of $3\Bbb Z$ are $n\Bbb Z$ for $n$ divisible by $3$. So it turns out $I$ is of the form $\left[\begin{smallmatrix}*&n\Bbb Z \\*&*\end{smallmatrix}\right]$ for some positive integer $n$ divisible by $3$, or else $n=0$.

It's elementary to show that if $\left[\begin{smallmatrix}0&n\\0&0\end{smallmatrix}\right]\in I$, then so are $\left[\begin{smallmatrix}n&0\\0&0\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}0&0\\0&n\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}0&0\\n&0\end{smallmatrix}\right]$. This establishes that $I\supseteq \left[\begin{smallmatrix}n\Bbb Z&n\Bbb Z\\n\Bbb Z&n\Bbb Z\end{smallmatrix}\right]$.

Hints

(based on the above notation)

If $n$ is a composite number (with nontrivial factorization $ab=n$, say), then look at the product of two ideals $\left[\begin{smallmatrix}a\Bbb Z&a\Bbb Z\\a\Bbb Z&a\Bbb Z\end{smallmatrix}\right]\left[\begin{smallmatrix}b\Bbb Z&b\Bbb Z\\b\Bbb Z&b\Bbb Z\end{smallmatrix}\right]\subseteq I$, and conclude $I$ can't be prime.

Now, any ideal having $3\Bbb Z$ in the upper right hand corner contains the ideal $A=\left[\begin{smallmatrix}3\Bbb Z&3\Bbb Z\\3\Bbb Z&3\Bbb Z\end{smallmatrix}\right]$. Look at the quotient by this ideal and use the correspondence theorem to count that there are five ideals in that quotient. Determine which ones are prime (because they correspond to prime ideals in $\Lambda$.)

Finally, you can show that if $n=0$, then $I=0$. (You will see that any nonzero entry has a nonzero multiple in the upper right hand corner.) The final step is to show that this ideal is prime too. I'd do this by proving that if $B$ and $C$ are nonzero elements of $\Lambda$, then there exists $D$ such that $BDC\neq 0$. Hint: do it in $M_2[\Bbb Q]$ first and then scale your $D$ into $\Lambda$.)