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Abstract: We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.

Let k be a field of characteristic 0. Based on the Gelfand–Kirillov dimension computation of modules over solvable polynomial k-algebras, where solvable polynomial algebras are in the sense of A. Kandri-Rody and V. Weispfenning (1990, J. Symbolic Comput.9, 1–26), we prove that the elimination lemma, obtained from D. Zeilberger (1990, J. Comput. Appl. Math.32, 321–368) by using holonomic modules over the Weyl algebra An(k) and used in the automatic proving of special function identities, holds for a class of solvable polynomial k-algebras without any “holonomicity” restriction. This opens a way to the solution of the extension/contraction problem stemming from the automatic proving of multivariate identities with respect to the ∂-finiteness in the sense of F. Chyzak and B. Salvy (1998, J. Symbolic Comput.26, 187–227). It also yields a ∂-holonomicity so that automatic proving of multivariate identities may be dealt with by manipulating polynomial function coefficients instead of rational functions.

Let S be a Krull domain which is divisorially étale of degree two over a Krull domain R. The main purpose of this text is to introduce and study involutive invariants of the second kind of S with respect to R, as well as developing techniques allowing to calculate these in terms of other, classical and involutive, invariants.

In this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the Hochschild cohomology of that coalgebra with coefficients in some one-dimensional bicomodules. As an application, for a given prime numberpand an algebraically closed fieldkof characteristic 0, we classify all pointed Hopf algebras of dimensionp3overk.

We apply to Hopf algebras a construction from graded rings, called “the group ring of a graded ring”. By using this construction we study the transfer of properties between certain categories of relative Hopf modules. As another application, we obtain a Maschke-type theorem for a Galois extension over a semisimple Hopf algebra.

The idea of a control subgroup or inertia group for objects over group graded rings is applied to filtratons of the objects defined over the graded ring. we stidy the interplay between properties of a foltered nature and induction from subgroups of the grading groups.

This paper generalizes properties which hold for localization of Azumaya algebras, in two directions. Firstly, fully left bounded left Noetherian rings, especially finitely generated Noetherian algebras, are considered. It is noted that for such rings every idempotent kernel functor a is symmetric, i.e. the filter T(σ) of a-dense left ideals has a basis of a-dense ideals. A prime ideal P of a f.l.b.l.N. ring R is localizable if and only if it is the intersection of the P-critical left ideals. In case R is a finitely generated algebra over its (Noetherian) center C, we apply the technique of "descent" of kernel functors. If a is a symmetric kernel functor such that R(A n c) S T(σ) for every A G T(σ) and such that a has property (T) then there is a kernel functor a' on C-modules such that Qσ (R) ≅Q≅ ,(R). If P is a prime ideal of R then σ- descends to C if and only if P is localizable. Secondly, a class of rings is described in terms of the Zariski topology on Spec. The imposed condition is weaker than maximal centrallity and does not imply fully left boundedness either, but the good properties of Spec R in case R is an Azumaya algebra are preserved.

So-called “quantized” algebras are popular objects of study in non-commutative algebra. Usually such algebras are either positively graded Ore domains R with R0 = K a field and R = K[R1], R1 being a finite dimensional K vectorspace, or else filtered rings having a ring of forementioned type for its associated graded ring. We show that every discrete valuation of K extends to a valuation, in the sence of O. Schilling (cf. [S]), of the skewfield of fractions, Δ = Qcl(R), of the Ore domain R (Proposition 2.3. and Corollary 2.8.). Such extension property has long been known to fail for finite dimensional skewfields over K; its validity in the case of several quantized algebras may be viewed as a consequence of the rigidity of their defining relations. Our result opens the door for a more arithmetical study of Δ e.g. in case K is a numberfield or an algebraic function field of a curve; for an application in this direction we refer to a first version of some divisor calculus started in [VW].