It is well-known that a ring R is semiperfect if and only if R R (or R R ) is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M ) is semisimple) and that R is a semilocal ring if and only if R R (or R R ) 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). I would like to thank Nyguen Viet Dung for bringing the R. Camps and W. Dicks paper to my attention. Moreover I want to express my thanks to Patrick F. Smith and Robert Wisbauer for their interest and helpful suggestions.

Abstract. 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 injective module admits a homconnection with respect to any differential graded algebra. The bulk of the article 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 calculus 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 calculus 1 .A/) that vanishes on the dual basis of 1 .A/. To any flat hom-connection r 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 r-integral. Examples of free twisted multi-derivations, hom-connections and corresponding integral forms are provided by covariant calculi on Hopf algebras (quantum groups). The example of a flat hom-connection within the 3D left-covariant differential calculus on the quantum group O q .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 Podleś sphere O q .S 2 / is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the r-integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.Mathematics Subject Classification (2010). 58B32; 16W25.

In a recent paper by Wang and Ding, it was stated that any ring which is generalized supplemented as a left module over itself is semiperfect. The purpose of this note is to show that Wang and Ding's claim is not true and that the class of generalized supplemented rings lies properly between the classes of semilocal and semiperfect rings. Moreover, we propose a corrected version of the theorem by introducing a wider notion of 'local' for submodules.2000 Mathematics subject classification: primary 16L30; secondary 16D10.

Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules L(M ) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which L(M ) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz' "weakly compressible" modules. In particular we are interested in representing weakly compressible modules as a subdirect product of "prime" modules in a suitable sense. It turns out that any weakly compressible module is a subdirect product of prime modules (in the sense of Kaplansky). Moreover if M is a self-projective module, then M is weakly compressible if and only if it is a subdirect product of prime modules (in the sense of Bican et al.). An application to Hopf actions is given.

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 for 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 Sweedler's 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. IntroductionAlfred Haar introduced a measure µ on the space of representable functions R(G) of a locally compact group G (see [Haa33]). The map f → f dµ is an integral I ∈ (R(G)) * . Hochschild exhibited the Hopf algebra structure of R(G) and characterised the G-invariance of I as I being a R(G)-colinear map (see [Hoc65,