Proving Maximal Ideal I in Noncommutative Ring R with Unity

[Fizz_Geek]

Question:

Find a noncommutative ring R with unity, with maximal ideal I such that R/I is not a field.

Attempt at a solution:

Let R = the set of all 2x2 matrices with integer entries.
Let I = the set of all 2x2 matrices with even integer entries.

I'm having trouble proving that I is maximal. The only way I know to do that is to assume I is not maximal, therefore it is contained within a maximal ideal J, then show that J = R. But I'm stuck. I haven't been able to prove that J = R.

Any help is appreciated!

 

[micromass]

Well, I don't know whether it is maximal either, but let's find out together. Denote $$M=M_2(2\mathbb{Z})$$. Assume that there exists an ideal $$M\subset I\subseteq M_2(\mathbb{Z})$$. Our goal is to show that $$I=M_2(\mathbb{Z})$$.

So, take an element $$A\in I\setminus M$$. Then we can assume without loss of generalization that

$$A=\left(\begin{array}{cc} 2a+1 & b\\ c & d\end{array}\right)$$

So, we assume that the first element is odd. We know nothing about b, c and d, they may be odd or even.

Now, the elementary matrix $$E_{i,j}$$ is such that there is a 1 on place (i,j) and a 0 on all other places. For example,

$$E_{2,1}=\left(\begin{array}{cc} 0 & 0\\ 1 & 0 \end{array}\right)$$.

Now the goal is to multiply A by suitable elementary matrices to simplify the form of A. For example, we might reduce A in the form

$$A=\left(\begin{array}{cc} 2a+1 & 0\\ 0 & 0\end{array}\right)$$

and since we obtained this form by multiplying by elementary matrices, this means that this must be an element of I. So, try to reuce the form of A into a suitable form. Once you've done this, we'll try to see what our next step is...

 

[Fizz_Geek]

So I did what you asked, and I multipled on the right and left by E_1,1. This gives the desired matrix. And I know this matrix is an element of I because I is an ideal and absorbs products.

The train of thought I was following started out the same, but I was trying to modify the given matrix so that I could prove that the identity matrix was contained in I. That's where I got stuck.

Can I ask where you're going with this?

 

[micromass]

So I did what you asked, and I multipled on the right and left by E_1,1. This gives the desired matrix. And I know this matrix is an element of I because I is an ideal and absorbs products.

The train of thought I was following started out the same, but I was trying to modify the given matrix so that I could prove that the identity matrix was contained in I. That's where I got stuck.

Can I ask where you're going with this?

We're also trying to show that the identity matrix is in I. The method with elementary matrices is a standard method of proving such a thing...

Now, try to prove that

$$\left(\begin{array}{cc}1 & 0\\ 0 & 0\end{array}\right)$$

is in I. Then try to multiply this matrix by a certain matrix to conclude that also

$$\left(\begin{array}{cc}0 & 0\\ 0 & 1\end{array}\right)$$

is in I. Adding these matrices together will get you the identity matrix!

 

[Fizz_Geek]

Oh, I got it!

You wouldn't happen to know any good resources with examples on this elementary matrix method, would you?

I really appreciate your help, thank you very much!

 

[micromass]

There is really nothing more that we can say about elementary matrices. It's just a trick that you know now. I think you know as much about elementary matrices as me right now...

But maybe you could read en.wikipedia.org/wiki/Elementary_matrix but it doesn't contain much more than you already know.