Ideals of a subring of $M_{2\times 2}(\mathbb{R})$ Define $$A:=\left[\begin{array}{cc}\mathbb{R}&\mathbb{R}\\ 0&\mathbb{R}\end{array}\right]=\left\{\left[\begin{array}{cc}a&b\\ 0&c\end{array}\right]:a,b,c\in\mathbb{R}\right\}$$ 
Prove that $A$ has exactly $5$ two sided ideals namely 
$$A, \left[\begin{array}{cc}\mathbb{R}&\mathbb{R}\\ 0&0\end{array}\right],\left[\begin{array}{cc}0&\mathbb{R}\\ 0&\mathbb{R}\end{array}\right],\left[\begin{array}{cc}0&\mathbb{R}\\ 0&0\end{array}\right],0$$
This was a recent exam problem. My solution though correct wasn't very nice. I checked that each of these was an ideal and then made a case by case arguement that any other two sided ideals must be one of these. I don't think that was the intended solution since all of the other problems on the exam had slick non computational solutions. 
I was wondering if anyone had any other ideas on how to solve this problem? Thanks in advance.
 A: Hint: What do you get if, using matrices $E_1:=\pmatrix{1&0\\0&0},\ E_2:=\pmatrix{0&1\\0&0},\ E_3:=\pmatrix{0&0\\0&1}\ \in A$ you form the 
$E_iME_j$ product for a matrix $M\in A$?
That will lead to the observation that e.g. if $\pmatrix{a&b\\0&0}$ is in an ideal $\mathcal I$, then both $\pmatrix{a&0\\0&0}$ and $\pmatrix{0&b\\0&0}$ must be in $\mathcal I$. If $a,b\ne 0$ then also $E_1,E_2\in\mathcal I$ hence $\pmatrix{\Bbb R&\Bbb R\\ 0&0 }\subseteq\mathcal I$ follows.
A: Notice that J(R) is the set of strictly upper triangular matrices, since it is a nilpotent ideal and modding it out makes a ring with Jacobson radical zero.
Moreover, you can easily prove that this ideal is contained in all other ideals. Since it's one dimensional, it's the unique minimal ideal of the ring.
The other ideals have to correspond to those in R/J(R), which is isomorphic to $R\times R$. The only ideals are the four obvious ones. Those, in addition to the original zero ideal furnish a complete list of two sided ideals.
