I'm currently reviewing ideals and I'm struggling specifically on how to express ideals of matrix rings. For instance, what would be the ideals of $M_2(\mathbb{Z}_6)$? I know that the ideals of $\mathbb{Z}_6$ are $(0),(1) = (5),(2) = (4)$, and $(3)$ and in general, if $I$ is an ideal of $R$ then $M_n(I)$ is an ideal of $M_n(R)$. Would it be valid to express the ideals of $M_2(\mathbb{Z}_6)$ as $M_2((0))$, $M_2((1))$, and so on?
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What you've said is correct so far. – rschwieb Dec 17 '15 at 04:42