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Let $R$ be the ring of matrices $$R=\big\{\begin{pmatrix} a & b\\ c & d \end{pmatrix} |a,b,c,d \in \mathbb{Z}_{2}\big\}$$

I want to find all the ideals of $R$. We know that $R$ has $16$ elements and therefore the possible orders of the non-trivial ideals are $2,4,8,16$. Besides taking all the possible cases for the elements of the matrices how can I approach this problem?

Teplotaxl
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    The ring $R=M_2(K)$ has only the trivial ideals, see this duplicate. Here $K=\Bbb Z/2$ is a field. – Dietrich Burde Jun 07 '22 at 13:38
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    It's true in general that the two-sided ideals of the ring of matrices over any field are ${0}$ and $R$. For the left ideals, there is a bijective correspondece between the vector subspaces of a vector space and the left ideals of its ring of matrices, given by $W\mapsto {A\in k^{n\times n},:, \ker A\supseteq W}$. Similarly, there is a bijective correspondence between the vector subspaces and the right ideals given by $W\mapsto {A\in k^{n\times n},:, \operatorname{col}A\subseteq W}$. – Sassatelli Giulio Jun 07 '22 at 13:41
  • For the right ideals, let $v_1,v_2$ be the two rows of $V$, then there are 5 cases for $V M_2$: $v_1K+v_2K=K^2$ and $ v_1=v_2=0$ and $v_1\ne 0,v_2=0$ and $v_2\ne 0,v_1=0$ and $v_1\ne 0,v_2 \in v_1 K^*$ – reuns Jun 07 '22 at 13:43

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