Submodule categories of wild representation type

Let ΛΛ be a commutative local uniserial ring with radical factor field kk. We consider the category S(Λ)S(Λ) of embeddings of all possible submodules of finitely generated ΛΛ-modules. In case Λ=Z/〈pn〉Λ=Z/〈pn〉, where pp is a prime, the problem of classifying the objects in S(Λ)S(Λ), up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that ΛΛ has Loewy length at least seven. We show that S(Λ)S(Λ) is controlled kk-wild with a single control object I∈S(Λ)I∈S(Λ). It follows that each finite dimensional kk-algebra can be realized as a quotient End(X)/End(X)IEnd(X)/End(X)I of the endomorphism ring of some object X∈S(Λ)X∈S(Λ) modulo the ideal End(X)IEnd(X)I of all maps which factor through a finite direct sum of copies of II.