# Form of Ideal of matrix over ring

Given $$R$$ be commutative ring and $$T =\begin{pmatrix} R & R \\ 0 & R \end{pmatrix} = \left\{ \left. \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \right| a,b,c \in R \right\}$$ be a ring of matrix over $$R$$.

In this paper https://www.researchgate.net/publication/312907984_On_Right_S-Noetherian_Rings_and_S-Noetherian_Modules on Example 1 said that if $$J$$ is an ideal of $$T$$, $$J$$ can be written as $$\begin{pmatrix} I_1 & I_2 \\ 0 & I_3 \end{pmatrix}$$ where $$I_1, I_2$$, and $$I_3$$ are ideals of $$R$$ satisfying $$I_1 \subseteq I_2$$.

I've read Proposition 1.17 in Lam T.Y, but I can't find any relation between the bold statement and those proposition.

Proposition 1.17. The right ideals of $$A$$ are the form $$J_1 \oplus J_2$$, where $$J_1$$ is right ideal in $$R$$, and $$J_2$$ is right $$S$$-submodule of $$M \oplus S$$ containing $$J_1M$$. Where $$R,S$$ be two rings, $$M$$ be an $$(R,S)$$-bimodule, and $$A= \begin{pmatrix} R & M \\ 0 & S \end{pmatrix} = \left\{ \left. \begin{pmatrix} r & m \\ 0 &s \end{pmatrix} \right| r \in R, m \in M, s \in S \right\}.$$

How to find the relation between the bold statement and those Proposition?

• Simply take $S = M = R$ in the Proposition 1.17. More specifically, first take $S = R$ (as a ring) and then take $M = R$ (as an $(R, R)$-bimodule). – WhatsUp Dec 27 '19 at 5:15
• Then why $I_1 \subseteq I_2$ ? – RANGGAJAYA CIPTAWAN Dec 27 '19 at 5:18
• You should translate the sentence "$J_2$ is right $S$-submodule of $M \oplus S$ containing $J_1M$". It's really straightforward... – WhatsUp Dec 27 '19 at 5:20
• So you take $J_1$ is $I_1$ and $J_2$ is $I_2$? – RANGGAJAYA CIPTAWAN Dec 27 '19 at 5:25
• $J_1$ is $I_1$ and $J_2$ is $I_2 \oplus I_3$. – WhatsUp Dec 27 '19 at 5:27

As mentioned in the comments, in your case, $$R=S=M$$ and $$I_1=J_1$$ and $$I_2\oplus I_3=J_2$$.
Suppose $$I$$ is an ideal of $$A$$, and we write $$E_{11}=\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$$ and $$E_{22}=\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$$. Then with $$I_1=E_{11}AE_{11}$$, $$I_2=E_{11}AE_{22}$$ and $$I_3=E_{22}AE_{22}$$, you can work to show that $$I_1\lhd R$$, $$I_3\lhd S$$ and $$I_2$$ is a sub-bimodule of $$M$$ containing both $$I_1M$$ and $$MI_3$$.