A module is both left artinian and left noetherian,but it is neither right artinian nor noetherian,

Question

I am reading Rings and Categories of Modules by Frank W.Aderson,on 130 pages. It ssys,the ring R of all 2$\times$2 upper triangular matrices \begin{Bmatrix} a & b \\ 0 & \gamma \end{Bmatrix}

with a,b$\in$R and $\gamma \in$Q is both left artinian and left noetherian,but it is neither right artinian nor right noetherian.

I can't prove it is not right artinian or right noetherian.

At the same time,I also can't undestand why it is both left artinian an right noetherian. Beacuse I even can't determine all submodule'form.For example thers are some obviously shape: \begin{Bmatrix} a_1 & b_1 \\ 0 & 0 \end{Bmatrix} \begin{Bmatrix} a_1 & 0 \\ 0 & 0 \end{Bmatrix}\begin{Bmatrix} 0 & b_1 \\ 0 & 0 \end{Bmatrix}\begin{Bmatrix} 0 & b_1 \\ 0 & \gamma \end{Bmatrix} but there are also other forms :\begin{Bmatrix} b_1 & b_1 \\ 0 & 0 \end{Bmatrix} So I can't deremine all the submodules'forms. I want someone can give me some help,thanks!

Answer 1

Let $U$ be a $\mathbb{Q}$-vector subspace of $\mathbb{R}$ and consider $$ I_U=\left\{\begin{bmatrix} 0 & x \\ 0 & 0 \end{bmatrix}:x\in U\right\} $$ This is obviously closed under addition. Moreover $$ \begin{bmatrix} 0 & x \\ 0 & 0 \end{bmatrix} \begin{bmatrix} a & b \\ 0 & \gamma \end{bmatrix} = \begin{bmatrix} 0 & x\gamma \\ 0 & 0\end{bmatrix}\in I_U $$ Hence $I_U$ is a right ideal.

Clearly, $U\subset U'$ implies $I_U\subset I_{U'}$.

Is $\mathbb{R}_{\mathbb{Q}}$ an artinian or noetherian module?