So I have shown in an earlier problem that the ideals of $M_2(\mathbb{R})$ are precisely the diagonal matrices. It seems to me that for the same reasons that those ideals needed to be diagonal, matrices with coefficients from $\mathbb{Z}_{12}$ will need to be diagonal. Furthermore, there must be additional conditions on which elements will work to generate the ideals, and my intuition is telling me that a matrix with diagonal coefficients of $d$ will be an ideal iff $gcd(d,n)=1$ or $gcd(d,n)\neq1$, but i'm having trouble making headway either way. Can somebody give me some advice? Thanks yall!
2026-03-29 10:15:28.1774779328
Find the ideals of $M_2(\mathbb{Z}_{12})$ and find their orders.
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