Christian Lomp

On semilocal modules and rings

It is well-known that a ring Ris semiperfect if and only if RR (orRR ) is a supplemented module. Considering weak supplementsinstead of supplements we show that weakly supplemented modules Mare semilocal (i.e.M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (orRR ) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal).

ON A RECENT GENERALIZATION OF SEMIPERFECT RINGS

It follows from a recent paper by Ding and Wang that any ring which is generalized supplemented as left module over itself is semiperfect. The purpose of this note is to show that Ding and Wangs claim is not true and that the class of generalized supplemented rings lies properly between the class of semilocal and semiperfect rings. Moreover we rectify their claim by introducing a wider notion of local submodules.

When is a smash product semiprime? A partial answer☆

Abstract It is an open question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H -invariant elements of the Martindale ring of quotients of a module algebra form a von Neumann regular and self-injective ring whenever A is semiprime. For a semiprime Goldie PI H -module algebra A with central invariants we show that A # H is semiprime if and only if the H -action can be extended to the classical ring of quotients of A if and only if every non-trivial H -stable ideal of A contains a non-zero H -invariant element. In the last section we show that the class of strongly semisimple Hopf algebras is closed under taking Drinfeld twists. Applying some recent results of Etingof and Gelaki we conclude that every semisimple cosemisimple triangular Hopf algebra over a field is strongly semisimple.

Integrals in Hopf Algebras over Rings

Abstract Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedlers equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.

Non-commutative integral forms and twisted multi-derivations

Non-commutative connections of the second type or hom-connections and associated integral forms are studied as generalisations of right connections of Manin. First, it is proven that the existence of hom-connections with respect to the universal differential graded algebra is tantamount to the injectivity, and that every finitely cogenerated injective module admits a hom-connection with respect to any differential graded algebra. The bulk of the paper is devoted to describing a method of constructing hom-connections from twisted multi-derivations. The notion of a free twisted multi-derivation is introduced and the induced first order differential calcu- lus is described. It is shown that any free twisted multi-derivation on an algebra A induces a unique hom-connection on A (with respect to the induced differential cal- culus 1 (A)) that vanishes on the dual basis of 1 (A). To any flat hom-connection ∇ on A one associates a chain complex, termed a complex of integral forms on A. The canonical cokernel morphism to the zeroth homology space is called a ∇-integral. Examples of free twisted multi-derivations, hom-connections and corresponding in- tegral forms are provided by covariant calculi on Hopf algebras (quantum groups). The example of a flat hom-connection within the 3D left-covariant differential cal- culus on the quantum group Oq(SL(2)) is described in full detail. A descent of hom-connections to the base algebra of a faithfully flat Hopf-Galois extension or a principal comodule algebra is studied. As an example, a hom-connection on the standard quantum Podles sphere Oq(S 2 ) is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the ∇- integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.

INJECTIVE MODULES OVER DOWN-UP ALGEBRAS

The purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian Down-Up algebras. We will show that the Noetherian Down-Up algebras A(\alpha,\beta,\gamma) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(\alpha,\beta,\gamma)-modules are locally Artinian provided the roots of X^2-\alpha X-\beta are distinct roots of unity or both equal to one.

Injective Hulls of Simple Modules over Differential Operator Rings

We study injective hulls of simple modules over differential operator rings R[θ; d], providing necessary conditions under which these modules are locally Artinian. As a consequence, we characterize Ore extensions of S = K[x][θ; σ, d] for σ a K-linear automorphism and d a K-linear σ-derivation of K[x] such that injective hulls of simple S-modules are locally Artinian.

Injective hulls of simple modules over finite dimensional nilpotent complex Lie superalgebras

Abstract We show that the finite dimensional nilpotent complex Lie superalgebras g whose injective hulls of simple U ( g ) -modules are locally Artinian are precisely those whose even part g 0 is isomorphic to a nilpotent Lie algebra with an abelian ideal of codimension 1 or to a direct product of an abelian Lie algebra and a certain 5-dimensional or a certain 6-dimensional nilpotent Lie algebra.

A central closure construction for certain algebra extensions. Applications to Hopf actions

Abstract Algebra extensions A ⊆ B where A is a left B -module such that the B -action extends the multiplication in A are ubiquitous. We encounter examples of such extensions in the study of group actions, group gradings or more general Hopf actions as well as in the study of the bimodule structure of an algebra. In this paper we are extending R.Wisbauers method of constructing the central closure of a semiprime algebra using its multiplication algebra to those kinds of algebra extensions. More precisely if A is a k -algebra and B some subalgebra of End ( A ) that contains the multiplication algebra of A , then the self-injective hull A ^ of A as B -module becomes a k -algebra provided A does not contain any nilpotent B -stable ideals. We show that under certain assumptions A ^ can be identified with a subalgebra of the Martindale quotient ring of A . This construction is then applied to Hopf module algebras.

ON THE NOTION OF STRONG IRREDUCIBILITY AND ITS DUAL

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories. 1. Irreducibility in semilattices 1.