I’m trying to work on the problem that asks to find all the ideals, (maximal and prime ideals) of $M_{2}(\mathbb{Z})$, the ring of all $2\times2$ matrices with integers entries.
So, I found that the set of all ideals of $M_{2}(\mathbb{Z})$ is essentially $M_{2}(n\mathbb{Z})$, where $n=0,1,...$. However, when I go on and find the prime and maximal ideals, I am not sure. Intuitively, I would say the $0$ matrix and $M_{2}(p\mathbb{Z})$, where $p$ is prime are prime ideals of $M_{2}(\mathbb{Z})$.
Are they actually prime ideals of this ring? If not, I would really appreciate any hint or suggestion.
One more thing to make sure is that, is $0$ a prime ideal? Since the definition state that a prime ideal has to be a proper ideal of the ring, which $0$ is indeed a proper ideal.
The answer is the same as this post, except that you can go further and say that the maximal ideals of $M_2(\mathbb Z)$ correspond to the maximal ideals of $\mathbb Z$. The prime ideals are of the form $M_2(\{0\})$ and $M_2(p\mathbb Z)$ for primes $p\in\mathbb Z$, and they're all maximal except for the former one.
Since you are talking about the prime ideals of a matrix ring, which is noncommutative, I assume you're using the standard definition of "prime ideal" for noncommutative rings.