# 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:

$$\left[\begin{array}{cc}\hfill S\hfill & \hfill S\hfill \\ \hfill 0\hfill & \hfill R\hfill \end{array}\right]:=\{\left[\begin{array}{cc}\hfill s\hfill & \hfill t\hfill \\ \hfill 0\hfill & \hfill r\hfill \end{array}\right]:s,t\in S,r\in R\}.$$ |

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.

$$R:=\left[\begin{array}{cc}\hfill \mathbb{R}\hfill & \hfill \mathbb{R}\hfill \\ \hfill 0\hfill & \hfill \mathbb{Q}\hfill \end{array}\right]=\{\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill 0\hfill & \hfill c\hfill \end{array}\right]:a,b\in \mathbb{R},c\in \mathbb{Q}\}.$$ | (1) |

###### Claim 2.

$R$ is left Artinian and left Noetherian^{}.

###### Proof.

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

Suppose that $z\ne 0$. Hence, ${s}_{q}:=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill q/z\hfill \end{array}\right]\in R$
for each $q\in \mathbb{Q}$
and so ${s}_{q}r=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill q\hfill \end{array}\right]\in I$ for all $q\in \mathbb{Q}$. In particular,
$\left[\begin{array}{cc}\hfill x\hfill & \hfill y\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]=r-{s}_{1}r\in I$. So in all cases it follows that
$\left[\begin{array}{cc}\hfill x\hfill & \hfill y\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\in I$. So now we take
$r=\left[\begin{array}{cc}\hfill x\hfill & \hfill y\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ 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 $\mathbb{R}$-vector space^{}, and so any
chain of left $R$-modules is a chain of subspaces^{}. As ${dim}_{\mathbb{R}}I\le 2$,
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 $3.14\mathrm{\dots}$), or any other transcendental number^{}, we define

$${I}_{n}:=\left[\begin{array}{cc}\hfill 0\hfill & \hfill \mathbb{Q}[\pi ;n]\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],$$ | (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 |