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Problem 24 (A Matrix Factor Ring)
Consider the set of two by two matrices $R=M_2(\mathbb{Z})$ with integer coefficients. Let $A$ the set of matrices whose coefficients are multiples of 3.
- Show that $A$ is an ideal of $R$.
- Find three different elements of $R$ that are in the coset $\begin{bmatrix}5&-2\\7&9\end{bmatrix}+A$.
- How many distinct elements are in this factor ring? Justify your answer.
- Challenge: This factor ring is isomorphic to another matrix ring we have already seen. What is this ring? You don't need to prove your answer.
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