Search results for "Locally nilpotent"

Locally nilpotent derivations of rings with roots adjoined

2013

Associative rings whose adjoint semigroup is locally nilpotent

2001

The set of all elements of an associative ring R, not necessarily with a unit element, forms a semigroup R ad under the circle operation \({r\circ s}={r+s+rs}\) on R. The ring R is called radical if R ad is a group. It is proved that the semigroup R ad is nilpotent of class n (in sense of A. Mal'cev or B. H. Neumann and T. Taylor) if and only if the ring R is Lie-nilpotent of class n. This yields a positive answer to a question posed by A. Krasil'nikov and independently considered by D. Riley and V. Tasic. It is also shown that the adjoint group of a radical ring R is locally nilpotent if and only if R is locally Lie-nilpotent.

Rationally integrable vector fields and rational additive group actions

2016

International audience; We characterize rational actions of the additive group on algebraic varieties defined over a field of characteristic zero in terms of a suitable integrability property of their associated velocity vector fields. This extends the classical correspondence between regular actions of the additive group on affine algebraic varieties and the so-called locally nilpotent derivations of their coordinate rings. Our results lead in particular to a complete characterization of regular additive group actions on semi-affine varieties in terms of their associated vector fields. Among other applications, we review properties of the rational counterpart of the Makar-Limanov invariant…

On Associative Rings with Locally Nilpotent Adjoint Semigroup

2003

Abstract The set of all elements of an associative ring R, not necessarily with a unit element, forms a semigroup R ad under the circle operation r ∘ s = r + s + rs for all r, s in R. This semigroup is locally nilpotent if every finitely generated subsemigroup of R ad is nilpotent (in sense of A. I. Mal'cev or B. H. Neumann and T. Taylor). The ring R is locally Lie-nilpotent if every finitely generated subring of R is Lie-nilpotent. It is proved that R ad is a locally nilpotent semigroup if and only if R is a locally Lie-nilpotent ring.

Radical Rings with Soluble Adjoint Groups

2002

Abstract An associative ring R , not necessarily with an identity, is called radical if it coincides with its Jacobson radical, which means that the set of all elements of R forms a group denoted by R ∘ under the circle operation r  ∘  s  =  r  +  s  +  rs on R . It is proved that every radical ring R whose adjoint group R ∘ is soluble must be Lie-soluble. Moreover, if the commutator factor group of R ∘ has finite torsion-free rank, then R is locally nilpotent.

Locally nilpotent derivations of rings graded by an abelian group

2019

International audience

Embeddings of Danielewski hypersurfaces

2008

In this thesis, we study a class of hypersurfaces in $\mathbb{C}^3$, called \emph{Danielewski hypersurfaces}. This means hypersurfaces $X_{Q,n}$ defined by an equation of the form $x^ny=Q(x,z)$ with $n\in\mathbb{N}_{\geq1}$ and $\deg_z(Q(x,z))\geq2$. We give their complete classification, up to isomorphism, and up to equivalence via an automorphism of $\mathbb{C}^3$. In order to do that, we introduce the notion of standard form and show that every Danielewski hypersurface is isomorphic (by an algorithmic procedure) to a Danielewski hypersurface in standard form. This terminology is relevant since every isomorphism between two standard forms can be extended to an automorphism of the ambiant …