Purely Inseparable Extensions of Factorial Rings

Hopf Galois structures on primitive purely inseparable extensions

Let L/K be a primitive purely inseparable extension of fields of characteristic p, [L : K] > p, p odd. It is well known that L/K is Hopf Galois for some Hopf algebra H, and it is suspected that L/K is Hopf Galois for numerous choices of H. We construct a family of K-Hopf algebras H for which L is an H-Galois object. For some choices of K we will exhibit an infinite number of such H. We provide ...

Scaffolds and integral Hopf Galois module structure on purely inseparable extensions

Let p be prime. Let L/K be a finite, totally ramified, purely inseparable extension of local fields, [L : K] = p, n ≥ 2. It is known that L/K is Hopf Galois for numerous Hopf algebras H, each of which can act on the extension in numerous ways. For a certain collection of such H we construct “Hopf Galois scaffolds” which allow us to obtain a Hopf analogue to the Normal Basis Theorem for L/K. The...

Extensions of Nilpotent P.P. Rings

Extensions of strongly alpha-reversible rings

We introduce the notion ofstrongly $alpha$-reversible rings which is a strong version of$alpha$-reversible rings, and investigate its properties. We firstgive an example to show that strongly reversible rings need not bestrongly $alpha$-reversible. We next argue about the strong$alpha$-reversibility of some kinds of extensions. A number ofproperties of this version are established. It is shown ...

Ore extensions of skew $pi$-Armendariz rings

For a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-Armendariz ring, that is a generalization of both $pi$-Armendariz rings, and $(alpha,delta)$-compatible skew Armendariz rings. We first observe the basic properties of skew $pi$-Armendariz rings, and extend the class of skew $pi$-Armendariz rings through various ring extensions. We nex...