|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'500'096
Articles rated: 2609
18 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
On pairs of matrices generating matrix rings and their presentations | | B.V. Petrenko
; S.N. Sidki
; | | Date: |
9 Dec 2005 | | Subject: | Rings and Algebras; Representation Theory | | Abstract: | Let $M_n(mathbb{Z})$ the ring of $n$-by-$n$ matrices with integral entries, and $n geq 2$. This paper studies the set $G_n(mathbb{Z})$ of pairs $(A,B) in M_n(mathbb{Z})^2$ generating $M_n(mathbb{Z})$ as a ring. We use several presentations of $M_{n}(mathbb{Z})$ with generators $X=sum_{i=1}^n E_{i+1,i}$ and $Y=E_{11}$ to obtain the following consequences. egin{enumerate} item Let $k geq 1$. Then the rings $M_n(mathbb{Q})^k$ and $igoplus_{j=1}^{k} M_{n_j} (mathbb{Z})$, where $n_1, ..., n_k geq 2$ are pairwise relatively prime, have presentations with 2 generators and finitely many relations.
item Let $D$ be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free. We use 4 relations for $X$ and $Y$ to describe all representations of the ring $M_{n}(D)$ into $M_{m}(D)$ for $m geq n$.
item We obtain information about the asymptotic density of $G_n(F)$ in $M_n(F)^2$ over different fields, and over the integers. end{enumerate} | | Source: | arXiv, math/0512186 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER