hollow matrix rings

1 Definition

Definition 1.

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

It is easy to check that this forms a ring under the usual matrix addition andmultiplication.This definition is slightly simplified from the obvious higher dimensional examplesand 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 idealstructure need not relate to its right ideal structure. We highlight a fewexamples of this.

2 Left/Right Artinian and Noetherian

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

(1)

Claim 2.

$R$ is left Artinian and left Noetherian.

Proof.

Let $I$ be a left ideal of $R$ and supposethat for some$x,y\in ℝ$ and $z\in \mathbb{Q}$.

Suppose that $z\ne 0$. Hence, for each $q\in \mathbb{Q}$and so for all $q\in \mathbb{Q}$. In particular,. So in all cases it follows that. So now we take and assume that $I$ does not contain any $r$ with $z\ne 0$.By observing matrix multiplication it follows that $I$ is now a left $ℝ$-vector space, and so anychain of left $R$-modules is a chain of subspaces. As ${dim}_{ℝ}I\le 2$,it follows that such chains are finite.

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

Claim 3.

$R$ is not right Artinian nor right Noetherian.

Proof.

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

(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 properascending and descending chain of right ideals. Therefore, $R$ is neither right Artiniannor right Noetherian.∎

Corollary 4.

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

Proof.

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