Q:

Infinite family of left ideals $J$ in a matrix ring $R$ and their direct sum.

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.

A:

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 $R\begin{bmatrix}0&0\\p&1\end{bmatrix}=\left\{\begin{bmatrix}0&0\\np&n\end{bmatrix}\,\middle|\, n\in\mathbb Z\right\}$.

