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Semiprime rings with nilpotent Lie ring of inner derivations

Kamil Kular (2014)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

We give an elementary and self-contained proof of the theorem which says that for a semiprime ring commutativity, Lie-nilpotency, and nilpotency of the Lie ring of inner derivations are equivalent conditions

Skew derivations and the nil and prime radicals

Jeffrey Bergen, Piotr Grzeszczuk (2012)

Colloquium Mathematicae

We examine when the nil and prime radicals of an algebra are stable under q-skew σ-derivations. We provide an example which shows that even if q is not a root of 1 or if δ and σ commute in characteristic 0, then the nil and prime radicals need not be δ-stable. However, when certain finiteness conditions are placed on δ or σ, then the nil and prime radicals are δ-stable.

Skew inverse power series rings over a ring with projective socle

Kamal Paykan (2017)

Czechoslovak Mathematical Journal

A ring R is called a right PS -ring if its socle, Soc ( R R ) , is projective. Nicholson and Watters have shown that if R is a right PS -ring, then so are the polynomial ring R [ x ] and power series ring R [ [ x ] ] . In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R [ [ x - 1 ; α , δ ] ] and the skew polynomial ring R [ x ; α , δ ] , where R is an associative ring equipped with an automorphism α and an α -derivation δ . Our results extend and unify many existing results....

Smooth invariants and ω -graded modules over k [ X ]

Fred Richman (2000)

Commentationes Mathematicae Universitatis Carolinae

It is shown that every ω -graded module over k [ X ] is a direct sum of cyclics. The invariants for such modules are exactly the smooth invariants of valuated abelian p -groups.

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