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.

$R$ is left Artinian and left Noetherian.

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
Canonical name	HollowMatrixRings
Date of creation	2013-03-22 17:42:21
Last modified on	2013-03-22 17:42:21
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	10
Author	Algeboy (12884)
Entry type	Example
Classification	msc 16P20
Classification	msc 16W10
Related topic	InvolutaryRing
Related topic	Artinian
Related topic	Noetherian2