The Endomorphism Ring Theorem for Galois and D2 extensions

对 Leroy 和 Matczuk 关于具有大像的半素左 Noetherian 环自同态单射性问题的肯定回答

Affirmative answer to the Question of Leroy and Matczuk on injectivity of endomorphisms of semiprime left Noetherian rings with large images

本文对 Leroy 和 Matczuk 提出的关于具有大像的半素左 Noetherian 环自同态是否必为单射的问题给出了肯定回答，并证明了更一般的二分法定理及具有 Krull 维数的半素环的相关结论。

Hopf-Galois structures on ambiskew polynomial rings

We provide necessary and sufficient conditions to extend the Hopf-Galois algebra structure on an algebra R to a generalized ambiskew ring based on R, in a way such that the added variables for the extension are skew-primitive in an appropriate sense. We show that the associated Hopf algebra is again a a generalized ambiskew ring, based on a suitable Hopf algebra H(R). Several examples are examined, including the Hopf-Galois objects over Uq(sl2).

A note on Galois theory for bialgebroids

In this note we reduce certain proofs in \cite{KS, Karl, AMA} to depth two quasibases from one side only. This minimalistic approach leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property: a proper algebra extension is a left $T$-Galois extension for some right finite projective left bialgebroid $T$ over some algebra $R$ if and only if it is of left depth two and left balanced. Exchanging left and right in this statement, we have also a characterization of right Galois extensions for left finite projective right bialgebroids…

Depth Two and the Galois Coring

We study the cyclic module ${}_SR$ for a ring extension $A \| B$ with centralizer $R$ and bimodule endomorphism ring $S = End {}_BA_B$. We show that if $A \| B$ is an H-separable Hopf subalgebra, then $B$ is a normal Hopf subalgebra of $A$. We observe from math.RA/0107064 and math.RA/0108067 depth two in the role of noncommutative normality (as in field theory) in a depth two separable Frobenius characterization of irreducible semisimple-Hopf-Galois extensions. We prove that a depth two extension has a Galois $A$-coring structure on $A ø_R T$ where $T$ is the right $R$-bialgebroid dual to $S$.

Normal Hopf subalgebras, depth two and Galois extensions

Let $S$ be the left $R$-bialgebroid of a depth two extension with centralizer $R$ as defined in math.QA/0108067. We show that the left endomorphism ring of depth two extension, not necessarily balanced, is a left $S$-Galois extension of $A^{\rm op}$. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We find a class of examples of the alternative Hopf algebroids in math.QA/0302325. We also characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are…

The Galois theory of matrix $C$-rings

A theory of monoids in the category of bicomodules of a coalgebra $C$ or $C$-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a $C$-ring (termed a {\em matrix $C$-ring}) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a {\em Galois module} is introduced and the structure theorem, generalising Schneider's Theorem II [H.-J. Schn…