# hollow matrix rings

## 1 Definition

###### Definition 1.

Suppose that $R\subseteq S$ are both rings. The hollow matrix ring of $(R,S)$ is the ring of matrices:

 $\begin{bmatrix}S&S\\ 0&R\end{bmatrix}:=\left\{\begin{bmatrix}s&t\\ 0&r\end{bmatrix}:s,t\in S,r\in R\right\}.$

It is easy to check that this forms a ring under the usual matrix addition and multiplication. This definition is slightly simplified from the obvious higher dimensional examples and the transpose of these matrices will also qualify as a hollow matrix ring.

The hollow matrix rings are highly counter-intuitive despite their simple definition. In particular, they can be used to prove that in general a ring’s left ideal structure need not relate to its right ideal structure. We highlight a few examples of this.

## 2 Left/Right Artinian and Noetherian

We specialize to an example with the fields $\mathbb{Q}$ and $\mathbb{R}$, though the same argument can be made in much more general settings.

 $R:=\begin{bmatrix}\mathbb{R}&\mathbb{R}\\ 0&\mathbb{Q}\end{bmatrix}=\left\{\begin{bmatrix}a&b\\ 0&c\end{bmatrix}:a,b\in\mathbb{R},c\in\mathbb{Q}\right\}.$ (1)
###### Claim 2.

$R$

###### Proof.

Let $I$ be a left ideal of $R$ and suppose that $r:=\begin{bmatrix}x&y\\ 0&z\end{bmatrix}\in I$ for some $x,y\in\mathbb{R}$ and $z\in\mathbb{Q}$.

Suppose that $z\neq 0$. Hence, $s_{q}:=\begin{bmatrix}0&0\\ 0&q/z\end{bmatrix}\in R$ for each $q\in\mathbb{Q}$ and so $s_{q}r=\begin{bmatrix}0&0\\ 0&q\end{bmatrix}\in I$ for all $q\in\mathbb{Q}$. In particular, $\begin{bmatrix}x&y\\ 0&0\end{bmatrix}=r-s_{1}r\in I$. So in all cases it follows that $\begin{bmatrix}x&y\\ 0&0\end{bmatrix}\in I$. So now we take $r=\begin{bmatrix}x&y\\ 0&0\end{bmatrix}$ and assume that $I$ does not contain any $r$ with $z\neq 0$. By observing matrix multiplication it follows that $I$ is now a left $\mathbb{R}$-vector space, and so any chain of left $R$-modules is a chain of subspaces. As $\dim_{\mathbb{R}}I\leq 2$, it follows that such chains are finite.

Hence, there can be no infinite descending chain of distinct left ideals and so $R$ is left Artinian and Noetherian. ∎

###### Claim 3.

$R$ is not right Artinian nor right Noetherian.

###### Proof.

Using $\pi$ (the usual $3.14\dots$), or any other transcendental number, we define

 $I_{n}:=\begin{bmatrix}0&\mathbb{Q}[\pi;n]\\ 0&0\end{bmatrix},$ (2)

where

 $\mathbb{Q}[\pi;n]:=\{q(\pi)\pi^{n}:q(x)\in\mathbb{Q}[x]\}.$ (3)

Since $\mathbb{Q}[\pi;n]$ properly contains $\mathbb{Q}[\pi;n+1]$ for all $n\in\mathbb{Z}$, it follows that $\{I_{n}:n\in\mathbb{Z}\}$ is an infinite proper ascending and descending chain of right ideals. Therefore, $R$ is neither right Artinian nor right Noetherian. ∎

###### Corollary 4.

$R$ does not have a ring anti-isomorphism. Thus $R$ is not an involutory ring.

###### Proof.

If $R$ is a ring with an anti-isomorphism, then the set of left ideals is mapped to the set of right ideals, bijectively and order preserving. This is not possible with $R$. ∎

Title hollow matrix rings HollowMatrixRings 2013-03-22 17:42:21 2013-03-22 17:42:21 Algeboy (12884) Algeboy (12884) 10 Algeboy (12884) Example msc 16P20 msc 16W10 InvolutaryRing Artinian Noetherian2