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On Engel groups and adjoint groups of nil rings | | Agata Smoktunowicz
; | | Date: |
1 Sep 2015 | | Abstract: | We show that there exists a nil ring R whose adjoint group $R^{o}$ is not an
Engel group. This answers a question by Amberg and Sysak from 1997 [5], and
also answers related questions from [3, 30]. We also show that over an
arbitrary uncountable field there is a nil algebra R whose adjoint group
$R^{o}$ is not an Engel group. This answers a recent question by Zelmanov.
In [25], Rump showed some surprising connections between Jacobson radical
rings and solutions to the Young-Baxter equation, and proved that every
Jacobson radical ring yields a solution to the Young-Baxter equation. In
particular, he showed that Jacobson radical rings are in one-to-one
correspondence with two-sided braces. In this context, Cedo, Jespers and
Okninski [13, 12] asked which groups are multiplicative groups of braces. A
similar question in the language of ring theory was asked in [4, 5].
We obtain some results related to these questions in Corollaries 3 and 4. It
is also worth noticing that, by a result by Gateva-Ivanova, braces are in
one-to-one orrespondence with braided groups with involutive Young-Baxter
operators [15]. | | Source: | arXiv, 1509.0420 | | Services: | Forum | Review | PDF | Favorites | |
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