The problem of the existence of sufficiently many injective Frechet modules over non-normed Frechet algebras

The main aim of this paper is to show that in the category of Frechet modules over certain Frechet algebras there cannot exist sufficiently many injective objects. In particular, we show that over Frechet algebras of formal power series there are no non-zero injective Frechet modules. We describe a class of Frechet algebras, which includes algebras of holomorphic functions over irreducible Stein spaces, over which there is no injective metrizable hypermodule. We also study the property of divisibility for Frechet modules and its relationship with the property of injectivity. We also show that every separable divisible Frechet module has periodic elements and prove a theorem on the non-existence of divisible Banach modules.