-class groups of certain extensions of degree

Let be an odd prime number. In this article we study the distribution of -class groups of cyclic number fields of degree , and of cyclic extensions of degree of an imaginary quadratic field whose class number is coprime to . We formulate a heuristic principle predicting the distribution of the -class groups as Galois modules, which is analogous to the Cohen-Lenstra heuristics concerning the prime-to--part of the class group, although in our case we have to fix the number of primes that ramify in the extensions considered. Using results of Gerth we are able to prove part of this conjecture. Furthermore, we present some numerical evidence for the conjecture.

     

A variant of the Reynolds operator

Let be a linearly reductive group over a field , and let be a -algebra with a rational action of . Given rational --modules and , we define for the induced -action on Hom a generalized Reynolds operator, which exists even if the action on Hom is not rational. Given an -module homomorphism , it produces, in a natural way, an -module homomorphism which is -equivariant. We use this generalized Reynolds operator to study properties of rational - modules. In particular, we prove that if is invariantly generated (i.e. ), then is a projective (resp. flat) -module provided that is a projective (resp. flat) -module. We also give a criterion whether an -projective (or -flat) rational --module is extended from an -module.

     

Limits of Pure-Injective Cotilting Modules

The set of pure-injective cotilting modules over an artin algebra is shown to have a monoid structure. This monoid structure does not restrict down to a monoid structure on the finitely generated cotilting modules in general, but it does whenever the algebra is of finite representation type. Pure-injective cotilting modules are also constructed from any set of finitely generated cotilting modules with bounded injective dimension. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G10, 16P20, 16E30.     

Identification of functional modules in a PPI network by clique percolation clustering

Large-scale experiments and data integration have provided the opportunity to systematically analyze and comprehensively understand the topology of biological networks and biochemical processes in cells. Modular architecture which encompasses groups of genes/proteins involved in elementary biological functional units is a basic form of the organization of interacting proteins. Here we apply a graph clustering algorithm based on clique percolation clustering to detect overlapping network modules of a protein–protein interaction (PPI) network. Our analysis of the yeast Sacchromyces cerevisiae suggests that most of the detected modules correspond to one or more experimentally functional modules and half of these annotated modules match well with experimentally determined protein complexes. Our method of analysis can of course be applied to protein–protein interaction data for any species and even other biological networks.     

C4- and D4-Modules via perspective direct summands

In this article, we study the C4- and D4-modules in terms of perspective direct summands, providing new characterizations and results. Endomorphism rings of C4-modules and extensions of right C4-rings are also investigated. Decompositions of C4-modules with restricted ACC on direct summands and D4-modules with restricted DCC on direct summands are obtained.     

On graded nil clean rings

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.     

Rings whose cyclic modules are lifting and ⊕-supplemented

It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules.     

Endopermutation Scott modules,slash indecomposability and saturated fusion systems

This paper introduces the notion of Frobenius-friendly modules which is a generalisation of p-permutation modules and we obtain the slash functors for these modules. Generalisations of Scott modules and Brauer indecomposability, which are called endopermutation Scott modules and slash indecomposability respectively, are given by using the slash functors. We also describe slash indecomposability in terms of saturated fusion systems, which is a generalisation of the case of Brauer indecomposability.