# Find ideals

Let $R=M_{2x2}$ of all rational numbers. Find all ideals.
The two obvious ones are $\{ 0 \}$ the trivial ideal and $M_2(\mathbb{Q})$ the improper one. However for a subring to be an ideal we require that the left multiplication is again constained in the subring in that case the last coloum is 0, and for right multiplication we require that the last row is 0. So the only such matrix which satisfies these properties is:
$\left( \begin{array}{cc}a&0\\0&0 \end{array} \right)$