Suppose I am given the ring, with basis elements $(e_1, e_2, \alpha, \beta)$.
$e_1$ and $e_2$ are idemponents $e_1^2 = e_1, e_2^2= e_2$ and the following holds: $$ \alpha e_1 = \alpha = e_2 \alpha \quad e_1 \alpha = 0 = \alpha e_2 $$ $$ \beta e_1 = 0 = e_2 \beta \quad e_1 \beta = \beta = \beta e_2 $$
$\alpha$ and $\beta$ are nilpotent : $\alpha^2 = \beta^2 = 0$
Therefore, all possible elements will have the form: $$ e_1, e_2, (\beta \alpha)^{k}, (\alpha \beta)^{k} , \alpha (\beta \alpha)^{k}, \beta (\alpha \beta)^{k} $$
By definition right ideal $I$ of a ring $R$ is the subgroup, such that $\forall r \in R \quad \forall i \in I$:
$$
i r \in I
$$
And maximal ideal is that, the addition of any element would enlarge it to all ring $R$.
For me it seems, that there are two maximal ideals, which consist of all elements, that end either on $\alpha$ or $\beta$. Multiplying by anything from the right we would still get something, that ends on $\alpha(\beta)$. The addition of something to the ideal, ending on $\alpha$ or $e_1, e_2$ would give the whole ring. Correct me, if I am wrong.
But doing this by guess and check seems not to be a good strategy. Is there a more clever way find maximal ideals?