a b s t r a c tWe construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The algorithm uses the root system of a complex simple Lie algebra. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka-Volterra system. We present in detail all such systems in the cases of A 3 , A 4 and we also give some examples from higher dimensions. We classify all possible Lotka-Volterra systems that arise via this algorithm in the A n case.

In this article we introduce two new concepts, those of a τ -CS and a s-τ -CS module, which are both torsion-theoretic analogues of CS modules. We investigate their relationship with the familiar concepts of τ -injective, τ -simple and τ -uniform modules and compare them with τ -complemented (τ -injective) modules, which were considered by other authors as torsiontheoretic analogues of CS modules. We are interested in decomposing a relatively CS module into indecomposable submodules, and in determining when a direct sum of relatively CS modules is relatively CS.

Topics in torsion theory STELIOS CHARALAMBIDESThe purpose of this thesis is to generalize to the torsion-theoretic setting various concepts and results from the theory of rings and modules. In order to accomplish this we begin with some preliminaries which introduce the main ideas used in torsion theory, the major ones being r-torsion and r-torsionfree modules as well as r-dense and r-pure submodules. In the first chapter we also introduce a new concept, that of a r-compact module, which is basic enough to deserve a place among the preliminaries.The results that we obtain fall into three areas which are to a certain degree interrelated. The first area is on r-Max modules, which we introduce as a torsion-theoretic analogue of Max modules. The main aim is to generalize a well-known result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsion-theoretic counterparts which we utilize in our generalization. Furthermore, we define and characterize left r-Max rings and apply the torsion-theoretic version of Shock's theorem to obtain a characterization of r-short modules motivated by a recent article in which short modules were introduced.The second area deals with various flavours of r-injectivity, some known and some new. We introduce r-Al-injective and s-r-A'l-injective modules and examine their relationship with the known concepts of r-injective and r-quasi-injective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s-r-AI-injective modules, and give a generalization of Azumaya's Lemma. We also prove that every M-generated module has a r-Al-injective hull which is unique up to isomorphism and show how this is linked to the r-quasi-injective hull. We then examine E-r-injectivity, generalizing well-known results by Faith, Albu and Nastasescu and Cailleau which provide necessary and sufficient conditions for the E-s-r-injective property, the E-s-r-A'l-injective property and for a direct sum of E-s-r-A4-injective modules to be E-s-r-A'f-injective.In the third area we introduce some new concepts with the aim of bringing to the torsion-theoretic setting the concept of a CS or extending module. The approach is twofold. The first is via r-CS modules which serve as a generalization of CS modules

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