ABELIAN GROUPS WITH A p²-BOUNDED SUBGROUP, REVISITED

Abstract

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be 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 p m A = 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 S 2 (Λ)/N Λ and S 2 (∆)/N ∆ are equivalent for certain nilpotent categorical ideals N Λ 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 2 -bounded for a given prime number p. determined uniquely by the lengths and m of the Λ-modules B and A; we write P m = (B; A).Clearly, the complexity of categories of type S m (Λ) increases with m. The categories S 0 (Λ) and mod Λ are equivalent so the only indecomposable objects in S 0 (Λ) are the pickets of type P 0 . In the category S 1 (Λ) (which we consider briefly in Sec. 3), every indecomposable object is a picket of type P 0 or P 1 . The category S 2 (Λ) contains additional indecomposables which are not pickets; it turns out that an invariant which has been introduced by Prüfer [3, §7] in 1923 provides an efficient classification. Definition. LetB be a Λ-module and a ∈ B be a nonzero element. The height exponent of a is h B (a) = h(a) = max{n ∈ N 0 : a = p n b for some b ∈ B}; the height sequence H B (a) = (h(a), h(pa), . . . , h(p a)) consists of the height exponents of the nonzero p-power multiples of a. Example 1.1. For pickets, the height sequence consists of consecutive numbers: In a picket (B; A) = P m where m > 0, any generator for A has the height sequence ( − m, − m + 1, . . . , − 1). J. Algebra Appl. 2011.10:377-389. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 06/21/15. For personal use only.