Let $R$ be a nil algebra over a field of characteristic 0, and let $\delta$ be a derivation of $R$. Then the differential polynomial ring $RX, \delta$ cannot be mapped onto a unital simple ring ...homomorphically.
Let R be an associative ring graded by left cancellative monoid S, and e the neutral element of S. We study the following problem: if Re is nil, then is R nil/nilpotent? We proved that if Re is nil ...(of bounded index) and f-commutative, then R is nil (of bounded index). Later, we showed that Re being nilpotent implies R is nilpotent. Consequently, we exhibited a generalization of Dubnov-Ivanov-Nagata-Higman Theorem for the graded algebras case. Furthermore, we exhibited relations between graded rings and the problems of Köthe and Kurosh-Levitzky. We proved that f-commutative rings provide positive solutions to these problems, and we also present a generalization of Kurosh-Levitzky Problem for the graded rings whose neutral components are f-commutative.
In this note we find the least order of a noncommutative even square ring and we note that it is a nil ring having characteristic four. In order to prove the main result given in this note we mainly ...use suitable examples.
In this paper, we study some connections between the polynomial ring
$Ry$
and the differential polynomial ring
$Rx;D$
. In particular, we answer a question posed by Smoktunowicz, which asks whether
...$Ry$
is nil when
$Rx;D$
is nil, provided that
$R$
is an algebra over a field of positive characteristic and
$D$
is a locally nilpotent derivation.
At the turn of the 21st century Agata Smoktunowicz constructed the first example of a nil algebra over a countable field such that the polynomial ring over the algebra is not nil. This answered an ...old question of Amitsur. We present a simplification of the example.
We discuss several problems on the structure of nil rings from the linear algebra point of view. Among others, a number of questions and results are presented concerning algebras of infinite matrices ...over nil algebras, and nil algebras of infinite matrices over fields, which are related to the famous Koethe's problem. Some questions on radicals of tensor products of algebras related to Koethe's problem are also discussed.
Problems on Skew Left Braces Leandro Vendramin
Advances in Group Theory and Applications,
06/2019, Volume:
7
Journal Article
Open access
Braces were introduced by Rump as a generalization of Jacobson radical rings. It turns out that braces allow us to use ring-theoretic and group-theoretic methods for studying involutive solutions to ...the Yang–Baxter equation. If braces are replaced by skew braces, then one can use similar methods for studying not necessarily involutive solutions. Here we collect problems related to (skew) braces and set-theoretic solutions to the Yang-Baxter equation.
The aim of this work is to study a decomposition theorem for rings satisfying either of the properties xy = x^pf(xyx)x^q or xy = x^pf(yxy)x^q, where p = p(x,y), q = q(x,y) are nonnegative integers ...and f(t) ∈ tZt vary with the pair of elements x, y, and further investigate the commutativity of such rings. Other related results are obtained for near-rings.
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