In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simple left mo...

In this paper we identify and explain diierent types of composition of speciications using defaults. Such defaults model overridable rules in object speciications. When composing object speciications it is desirable to control to what degree the intended behavior of the composition depends on the local intended behaviors. For this purpose two types of composition, open and closed composition, a...

Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simp...

We study classes of modules over a commutative ring which allow to do homological algebra relative to such a class. We classify those classes consisting of injective modules by certain subsets of ideals. When the ring is Noetherian the subsets are precisely the generization closed subsets of the spectrum of the ring.

In this topic proposal, we discuss the representation theory of semisimple Lie algebras, including results on the structure of these algebras and their modules. In particular, the theory of root systems and Verma modules are very important in this context. We classify all simple Lie algebras, and obtain a presentation for any semisimple Lie algebra. We also encounter several striking formulas t...

Let k be an algebraically closed field of characteristic p, possibly zero, and G = qGL3(k), the quantum group of three by three matrices as defined by Dipper and Donkin. We may also take G to be GL3(k). We first determine the extensions between simple G-modules for both G and G1, the first Frobneius kernel of G. We then determine the submodule structure of certain induced modules, Ẑ(λ), for the...

Let R be an associative ring with identity. It is shown that every Σ -cotorsion left R-module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ -cotorsion. It follows that, for countable R, the class of Σ -cotorsion modules is closed under elementary equivalence and pure submodules. The modules M t...

We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded a-invariant, and all modules with bounded Castelnuovo–Mumford regularity. The f...