Second order differential invariants of linear frames.
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
A ring or an idempotent semiring is associative provided that additive endomorphisms are multiplicative.
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.
A ring is called a right -ring if its socle, , is projective. Nicholson and Watters have shown that if is a right -ring, then so are the polynomial ring and power series ring . In this paper, it is proved that, under suitable conditions, if has a (flat) projective socle, then so does the skew inverse power series ring and the skew polynomial ring , where is an associative ring equipped with an automorphism and an -derivation . Our results extend and unify many existing results....
It is shown that every -graded module over is a direct sum of cyclics. The invariants for such modules are exactly the smooth invariants of valuated abelian -groups.