Each object-oriented programming language proposes various grouping mechanisms to bundle interacting classes (i.e., packages, modules, selector namespaces, etc). To understand this diversity and to compare the different approaches, a common foundation is needed. In this paper we present a simple module calculus consisting of a small set of operators over environments and modules. Using these op...

Let R be a regular ring essentially of finite type over a perfect field k. An R–module M is called a unit R[F ]–module if it comes equipped with an isomorphism F M −→ M where F denotes the Frobenius map on SpecR, and F e∗ is the associated pullback functor. It is well known that M then carries a natural DR–module structure. In this paper we investigate the relation between the unit R[F ]–struct...

Let R be a regular ring, essentially of finite type over a perfect field k. An R–module M is called a unit R[F ]–module if it comes equipped with an isomorphism F M −→ M, where F denotes the Frobenius map on SpecR, and F e∗ is the associated pullback functor. It is well known that M then carries a natural DR–module structure. In this paper we investigate the relation between the unit R[F ]–stru...

Preface These notes are based on a course given at the Tata Institute of Fundamental Research in the beginning of 1990. The aim of the course was to describe the solution by O. Mathieu of some conjectures in the representation theory of semi-simple algebraic groups. These conjectures concern the inner structure of dual Weyl modules and some of their analogues. Recall that if G is a (connected, ...

We make a first step towards a classification of simple generalized HarishChandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras (g, k), we construct, via cohomological induction, the fundamental series F ·(p, E) of generalized Harish-Chandra modules. We then use F ·(p, E) to characterize any simpl...

The current work describes an instructional module which emphasizes integrative design using the six simple machines of the lever, the wheel and axle, the pulley, the inclined plane, the screw, and gear. The emphasis of this module is on having the students investigate the underlying scientific and mathematical properties of the ‘machines’, and then integrating this knowledge to design creative...

Despite its powerful module system, ML has not yet evolved for the modern world of dynamic and open modular programming, to which more primitive languages have adapted better so far. We present the design and semantics of a simple yet expressive firstclass component system for ML. It provides dynamic linking in a type-safe and type-flexible manner, and allows selective execution in sandboxes. T...

Let R be a right GF -closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorenstein ring R, the G...

It is shown that the support of an irreducible weight module over the SchrödingerVirasoro Lie algebra with an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite-dimensional. As a side-product, it is obtained that every simple weight module over the Schrödinger-Virasoro Lie algebra with a nontrivial finite-dime...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...