in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

in this paper we introduce the concept of dirac structures on (hermitian) modules and vectorbundles and deduce some of their properties. among other things we prove that there is a one to onecorrespondence between the set of all dirac structures on a (hermitian) module and the group of allautomorphisms of the module. this correspondence enables us to represent dirac structures on (hermitian)mod...

Let  be a commutative ring and  be a unitary  module. We define a semi prime sub module of a module and consider various properties of it. Also we define semi-radical of a sub module of a module and give a number of its properties. We define modules which satisfy the semi-radical formula and present the existence of such a module.

Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and b 6= 1, α = b − 1. The condition on α implies that Γ is formally self-dual. For b = q we use the adjacency matrix and dual adjacency matrix to obtain an action of the q-tetrahedron algebra ⊠q on the standard module of Γ. We describe four algebra homomorphisms into ⊠q from the quantum affine algebra Uq(ŝl2); using t...

An action of the Yangian of the general linear Lie algebra glN is defined on every irreducible integrable highest weight module of ĝlN with level > 2. This action is derived, by means of the Drinfeld duality and a subsequent semi-infinite limit, from a certain induced representation of the degenerate double affine Hecke algebra H. Each vacuum module of ĝlN is decomposed into irreducible Yangian...

An action of the Yangian of the general linear Lie algebra glN is defined on every irreducible highest weight module of ĝlN with integer level > 2. This action is derived, by means of the Drinfeld duality and a subsequent semi-infinite limit, from a certain induced representation of the degenerate double affine Hecke algebra H. Each vacuum module of ĝlN is decomposed into irreducible Yangian re...

Let Uq(ŝl2) ≥0 denote the Borel subalgebra of the quantum affine algebra Uq(ŝl2). We show that the following hold for any choice of scalars ε0, ε1 from the set {1,−1}. (i) Let V be a finite-dimensional irreducible Uq(ŝl2) -module of type (ε0, ε1). Then the action of Uq(ŝl2) ≥0 on V extends uniquely to an action of Uq(ŝl2) on V . The resulting Uq(ŝl2)-module structure on V is irreducible and of ...

This paper, reports on the planning and preliminary results of an action research project undertaken for the redesign of an online distance learning information literacy (IL) module on the basis of virtual ‘embedded librarianship’. The research project, which followed an action research design, brought together the IL module coordinator and an Academic Liaison Librarian, working at different in...

We offer a reflection on our own praxis as designers and developers of a learning system for mature-age students through the Open University (OU) UK’s internationally recognised supported-open learning approach. The learning system (or course or module), which required an investment in the range of £0.25–0.5 million to develop, thus reflects our own history (traditions of understanding), the hi...

Action Selection schemes, when translated into precise algorithms, typically involve considerable design eeort and tuning of parameters. Little work has been done on solving the problem using learning. This paper compares eight diierent methods of solving the action selection problem using Reinforcement Learning (learning from rewards). The methods range from centralised and cooperative to dece...