hollow matrix rings

1 Definition

Definition 1.

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


It is easy to check that this forms a ring under the usual matrix additionMathworldPlanetmath and multiplication. This definition is slightly simplified from the obvious higher dimensional examples and the transposeMathworldPlanetmath 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 idealMathworldPlanetmath 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 and , though the same argument can be made in much more general settings.

R:=[0]={[ab0c]:a,b,c}. (1)
Claim 2.

Let I be a left ideal of R and suppose that r:=[xy0z]I for some x,y and z.

Suppose that z0. Hence, sq:=[000q/z]R for each q and so sqr=[000q]I for all q. In particular, [xy00]=r-s1rI. So in all cases it follows that [xy00]I. So now we take r=[xy00] and assume that I does not contain any r with z0. By observing matrix multiplication it follows that I is now a left -vector spaceMathworldPlanetmath, and so any chain of left R-modules is a chain of subspacesPlanetmathPlanetmathPlanetmath. As dimI2, 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.


Using π (the usual 3.14), or any other transcendental numberMathworldPlanetmath, we define

In:=[0[π;n]00], (2)


[π;n]:={q(π)πn:q(x)[x]}. (3)

Since [π;n] properly contains [π;n+1] for all n, it follows that {In:n} 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.


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