Abelian Groups with a P -bounded Subgroup, Revisited

Let Λ be a commutative local uniserial ring of length n, p a generator of the maximal ideal, and k the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆ B a submodule of B such that pmA = 0 form the objects in the category Sm(Λ). We show that in case m = 2 the categories Sm(Λ) are in fact quite similar to each other: If also ∆ is a commutative local uniserial ring of length n and with radical factor field k, then the categories S2(Λ)/NΛ and S2(∆)/N∆ are equivalent for certain nilpotent categorical ideals NΛ and N∆. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p-bounded for a given prime number p. 1. History and Introduction Let Λ be a commutative local uniserial ring of length n with radical generator p and radical factor field k = Λ/p. We consider pairs (B;A) where B is a finitely generated Λ-module andA a submodule ofB. Such pairs form the objects in the category S(Λ); a morphism from (B;A) to (D;C) is given by a map f : B → D which satisfies f(A) ⊂ C. We are particularly interested in the full subcategories Sm(Λ) ⊂ S(Λ) (for m ≤ n a natural number) which consist of those pairs (B;A) that satisfy pA = 0. For example if Λ = Z/p then we are dealing with pairs (B;A) where B is a finite abelian p-bounded group and A ⊆ B a subgroup satisfying pA = 0. Each category Sm(Λ) has the Krull-Remak-Schmidt property, so every object has a unique direct sum decomposition into indecomposable ones. Examples for indecomposable objects are pickets which are pairs (B;A) where the Λ-module B itself is indecomposable, hence cyclic. 2000 Mathematics Subject Classification : 20E07 (primary), 16G20 (secondary)