Locally convex modules

Let $K$ be a non-archimedean valued field, $A\subseteq K$ be its integer ring. This paper is devoted to the study of the locally convex topological unital $A$-modules. These modules are very close to the vector spaces over non-archimedean valued fields. In particular, the topology of these modules can be determined by some system $\Gamma$ of semipseudonorms. Monna demonstrated that $p$-adic analogue of Hahn–Banach theorem can be proved for the locally convex vector spaces over non-archimedean valued fields. One can give the definitions of $q$-injectivity, where $q$ is the seminorm which is determined on this module, and of the strong topological injectivity. It means that any $q$-bounded homomorphism can be extended with the same seminorm, where $q$ is a some fixed seminorm in the first case, and an arbitrary seminorm $q\in\Gamma$ in the second one. The necessary and sufficient conditions of $q$-injectivity and strong topological injectivity for torsion free modules are given. At last, the necessary and sufficient conditions for topological injectivity of a locally convex $A$-module in the case when $A$ is the integer ring of the main local compact non-archimedean valued field are the following ones: a topological module is complete and Baire condition holds for any continuous homomorphism (here topological injectivity means that any continuous homomorphism of a submodule can be extended to a continuous homomorphism of the whole module).