Ideal of upper triangular matrices over non noetherian ring.

Question

Let $T$ be a ring of upper triangular matrices over $R$, where $R$ is a commutative non Noetherian ring. Let $J$ be a right ideal of $T$. Why $J$ can be written as $$ \begin{bmatrix} I_1&I_2\\ 0&I_3 \end{bmatrix} $$ where $I_1,I_2,$ and $I_3$ are ideals of $R$ satisfying $I_1 \subseteq I_2$?

Answer 1

That is not a correct description.

The collection $\begin{bmatrix}0& x \\ 0&y\end{bmatrix}\in J$ only forms a right $R$ submodule of $\begin{bmatrix}0& R\\ 0&R\end{bmatrix}$, and it is not always possible to find two right ideals of $R$ such that the submodule is the sum like this.

You should take a look at this A first course in noncommutative rings by Lam - Proposition 1.17 about triangular rings.

It has nothing to do with the ring being non-Noetherian.