| Title:
|
Two remarks on Lie rings of $2\times 2$ matrices over commutative associative rings (English) |
| Author:
|
Bashkirov, Evgenii L. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
1-10 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $C$ be an associative commutative ring with 1. If $a\in C$, then $aC$ denotes the principal ideal generated by $a$. Let $l, m, n$ be nonzero elements of $C$ such that $mn \in lC$. The set of matrices $\left( \begin{smallmatrix} a_{11} & a_{12} a_{21} & -a_{11} \end{smallmatrix} \right) $, where $a_{11}\in lC$, $a_{12}\in mC$, $a_{21}\in nC$, forms a Lie ring under Lie multiplication and matrix addition. The paper studies properties of these Lie rings. (English) |
| Keyword:
|
Lie ring |
| Keyword:
|
associative commutative ring |
| Keyword:
|
matrix |
| MSC:
|
17B05 |
| idZBL:
|
Zbl 07217153 |
| idMR:
|
MR4093424 |
| DOI:
|
10.14712/1213-7243.2020.010 |
| . |
| Date available:
|
2020-04-30T11:11:31Z |
| Last updated:
|
2022-04-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148068 |
| . |
| Reference:
|
[1] Bashkirov E. L.: On a class of Lie rings of $2\times 2$ matrices over associative commutative rings.Linear Multilinear Algebra 67 (2019), no. 3, 456–478. MR 3909003, 10.1080/03081087.2017.1422235 |
| Reference:
|
[2] Bashkirov E. L., Pekönür E.: On matrix Lie rings over a commutative ring that contain the special linear Lie ring.Comment. Math. Univ. Carolin. 57 (2016), no. 1, 1–6. MR 3478334 |
| Reference:
|
[3] Borevich, A. I., Shafarevich I. R.: Number Theory.Pure and Applied Mathematics, 20, Academic Press, New York, 1966. MR 0195803 |
| Reference:
|
[4] Ireland K., Rosen M.: A Classical Introduction to Modern Number Theory.Graduate Texts in Mathematics, 84, Springer, New York, 1990. MR 1070716 |
| Reference:
|
[5] Koibaev V. A., Nuzhin Ya. N.: Subgroups of Chevalley groups and Lie rings of definable by a collection of additive subgroups of the original ring.Fundam. Prikl. Mat. 18 (2013), no. 1, 75–84 (Russian. English, Russian summary); translated in J. Math. Sci. (NY) 201 (2014), no. 4, 458–464. MR 3431766 |
| Reference:
|
[6] Nuzhin Ya. N.: Lie rings defined by the root system and family of additive subgroups of the initial ring.Proc. Steklov Inst. Math. 283 (2013), suppl. 1, S119–S125. MR 3476385 |
| . |
