This exercise is from a previous exam.
Let $R = \begin{bmatrix}\mathbb{Z} & 0 \\ \mathbb{Z} & \mathbb{Z}\end{bmatrix}$ and $I = \begin{bmatrix} \mathbb{Z} & 0 \\ \mathbb{Z} & 0\end{bmatrix}$ be a left ideal of $R$. Find an infinite number of left ideals $J$ in $R$ such that $R = I \oplus J$
Firstly, I don't know exactly what they're asking me to do. If I find an infinite number of left ideals $\{J_i\}_{i=1}^{\infty}$ such that $R = I \oplus (J_1 \oplus J_2 \oplus ...)$, would that be correct? Or is it more likely that I'm supposed to find $\{J_i\}_{i=1}^{\infty}$ such that $R = I \oplus J_i$ for all $i=1,2,...$?
I tried to construct the left ideals $J$ like this: $J_p = \begin{bmatrix} 0 & 0 \\ 0 &\mathbb{Z}p\end{bmatrix}$ for all the primes $p$, but then the infinite sum $I + J_2 + J_3 + J_5 + ...$ isn't direct, and the direct sum $I \oplus J_p \neq R$. I'm not sure how I should construct the $J$'s to solve this, and it doesn't help that I don't know exactly what I'm suposed to be solving.
I would very much appreciate some small hints on how to solve this, and which of the two direct sums you think is the one they're asking for.
You weren't far off. You can't get around having $\mathbb Z$ in the lower right hand corner though: after subtracting elements in $I$, you have to be able to get anything in the lower right corner.
Try ideals of the form $R\begin{bmatrix}0&0\\p&1\end{bmatrix}=\left\{\begin{bmatrix}0&0\\np&n\end{bmatrix}\,\middle|\, n\in\mathbb Z\right\}$.