I am looking for a book or review article or original article that discusses what is known and what are the difficulties of working with matrices over noncommutative rings and algebras. While I have some very specific things in mind, I would like to have some idea what is known in general. Any suggestions would be welcomed.
Thanks.
Do not know of a specific reference, but suppose it is easy to find a set of matrices which do not commute, they normally do not, and which are not
invertible, or not all invertible. But the meaning or utility of this would
mainly escape my understanding. Maybe if you find a finite set of these, with closure.
Or you mean matrix elements are rings? Matrices over whole numbers.
There are some uses there.
regards, Juan
Dear Juan Weisz ,
> Or you mean matrix elements are rings?
I mean matrices in which the entries are not from, say R or C, but much more general rings and in particular, noncommutative rings and/or algebras.
Found the following on the internet and hope they could be of help:
https://link.springer.com/article/10.1007%2FBF01079588
https://math.stackexchange.com/questions/1977800/matrices-over-noncommutative-rings
https://scholar.google.com/scholar?rlz=1C1CAFB_enSG782SG782&um=1&ie=UTF-8&lr&cites=10970270971723017733
Vikas Mishra Thanks.
The work on quasi-determinants by Gel'fand and Retakh I know.
Anyway, once again thanks.
That is really exotic, non commutative entities whose components are also noncommutative; this would deprive
me from sleep.
A-6. Matrix rings
Answer 6, posted on August 12, 2019.
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I have found the following pieces of information
on the following Web page:
---
Matrix ring
https://en.wikipedia.org/wiki/Matrix_ring
---
Definition. A matrix ring is a set S of n by n matrices
whose entries belong to a given ring R
such that the set S provided with addition
and multiplication of matrices forms a ring.
---
The ring R is not necessarily commutative.
---
Main reference:
---
Lectures on Modules and Rings
Tsit-Yuen Lam
Graduate Texts in Mathematics 189
DOI: 10.1007/978-1-4612-0525-8
Hardcover ISBN-13: 9780387984285
Softcover ISBN-13: 9781461268024
eBook ISBN-13: 9781461205258
557 + xxiii pages, Springer, 1999
https://www.springer.com/us/book/9780387984285
--------------------------------------------
With best regards, Jean-Claude
Jean-Claude Evard
That is the kind of objects I am interested in. Though, I now have a clear answer to the specific question I was forced to tackle.
Thanks.
A-8. Quaternionic matrices
Answer 8, posted on August 13, 2019.
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I thank Andrew very much for the answer 7.
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The following Web page contains useful
information on the subject of this thread:
---
Quaternionic matrices
https://en.wikipedia.org/wiki/Quaternionic_matrix
---
Notably, this Web page contains a reference
to the following wonderful article:
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Quaternionic Determinants
Helmer Aslaksen
The Mathematical Intelligencer
Volume 18, issue 3, pages 57--65
DOI: 10.1007/BF03024312
Helmer Aslaksen
June 1996
https://link.springer.com/article/10.1007/BF03024312
https://www.researchgate.net/publication/226528876_Quaternionic_Determinants
---
This article is written with outstanding clarity.
The content is very strong.
In addition, the author has accomplished
an enormous amount of successful work
on the history of the subject.
The bibliography contains 52 references
that are very difficult to find.
--------------------------------------------
With best regards, Jean-Claude
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