Separable extensions of noncommutative rings

Pseudo-Galois Extensions and Hopf Algebroids

“…[19, 2.4]. Another example of a pseudo-Galois extension is an H-separable extension A | B, which by definition satisfies A ⊗ B A ⊕ * ∼ = A N as A-A-bimodules, so we let G = {id A } in the definition above [7,8,17]. For example, if A is a simple ring, G a finite group of outer automorphisms of A such that each nonidentity automorphism moves an element of the center of A, then the skew group ring A ⋊ G is H-separable over A [18].…”

“…If e i,σ corresponds via the other isomorphism above with g i,σ , then g i,σ (a) = ae i,σ . We compute: (17) x ⊗ B y = i,σ∈G…”

“…an Azumaya algebra) and group-Galois extension (e.g. a Galois algebra) [17]. Its Galois bialgebroid is shown in Theorem 2.3 to be closely related to a certain Hopf algebroid (in the sense of Böhm and Szlachányi [3]) we obtain from a cofibered sum of a semi-direct product of an algebra with a group of automorphisms and its opposite right crossed product.…”

Codepth Two and Related Topics

“…We need to add a hypothesis in order to obtain some result, such as V | B is a separable extension. In preparation for the next theorem, we make a definition of a weakened notion of depth two, which is analogous to the weakening of H-separability in the notion of strong separability introduced in McMahon and Mewborn [14]. Definition 2.2.…”

“…In Section 2 of this paper we pursue a more general fact underlying this; that a depth two extension A | B where the double centralizer W = V A (V A (B)) is a separable extension of B has weak depth two extension A | W . The definition of weak depth two, given in section 2, is modelled on Mewborn and McMahon's weakening of H-separability to "strong separability" in [14]. In section 3 we consider counterexamples and propositions for other closure and transitivity properties of depth two in a tower of algebras A ⊃ C ⊃ B.…”