Find all maximal ideals of the ring $M=\left\{ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \mid a \in \Bbb Z, b \in \Bbb R \right\}$.

Question

My problem: $M=\left\{ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \mid a \in \Bbb Z, b \in \Bbb R \right\}$.

I need to find all ideal I of M: M/I is a field.

We know that: An ideal I of a unitary commutative ring R is maximal iff R/I is a field. But i don't know how to find all maximal ideal of M so i'm stuck here. Any help is appreciated.

Answer 1

The ring $M$ is isomorphic to the product ring $\mathbb{Z} \times \mathbb{R}$, so the problem is equivalent to finding the maximal ideals of $\mathbb{Z} \times \mathbb{R}$. Any ideal of $R \times S$ must be of the form $I \times J$ for some ideals $I$ of $R$ and $J$ of $S$, and $I \times J$ is maximal (or prime) iff either $I=R$ and $J$ is maximal (or prime), or $J=S$ and $I$ is maximal (or prime).

Hence, the maximal ideals of $\mathbb{Z} \times \mathbb{R}$ are $p\mathbb{Z} \times \mathbb{R}$ where $p$ is a prime number, and $\mathbb{Z} \times \{0\}$.

Transporting to the ring $M$, we see that its maximal ideals are the following:

• The ideal of real diagonal $2 \times 2$ matrices where the top-left entry is an integer multiple of some fixed prime number $p$.
• The ideal of $2 \times 2$ matrices where the top-left entry is an integer and the other three entries are all zero.