We describe the structure of finitely generated cotorsion modules over commutative noetherian rings. Also we characterize the so-called covering morphisms between finitely generated modules over these rings.

AMS 2000 Mathematics subject classification: Primary 16D10. Secondary 13C11

We prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonné, which is well known in the theory of locally convex spaces.

We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have called the quasi-convex compactness property, or briefly qcp (Section 3).

The aim of this paper is to explore some properties of quasiuniform multifunction spaces. Various kinds of completeness of the quasiuniform multifunction space (YmX, UmX) are characterized in terms of suitable properties of the range space (Y, U). We also discuss the local compactness of quasiuniform multifunction spaces. By using the notion of small-set symmetry, the classic result of Hunsaker and Naimpally is extended to the quasiuniform setting.

In an earlier paper we proved that a universal Horn class generated by finitely many finite structures has a model companion. If the language has only finitely many fundamental operations then the theory of the model companion admits a primitive recursive elimination of quantifiers and is primitive recursive. The theory of the model companion is ℵ0-categorical iff it is complete iff the universal Horn class has the joint embedding property iff the universal Horn class is generated by a single finite structure. In the last section we look at structure theorems for the model companions of universal Horn classes generated by functionally complete algebras, in particular for the cases of rings and groups.