
Find ideals
Let $\displaystyle R=M_{2x2}$ of all rational numbers. Find all ideals.
I understand that we are talking about the set of 2x2 matrices with all rational number entries. An ideal is a subring that absorbs all r in R. So would that be the set of 2x2 matrices with integer entries?

The two obvious ones are $\displaystyle \{ 0 \}$ the trivial ideal and $\displaystyle M_2(\mathbb{Q})$ 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:
$\displaystyle \left( \begin{array}{cc}a&0\\0&0 \end{array} \right)$