Abelian groups with a $p^2$-bounded subgroup, revisited

Abstract

Let $R$ 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 $R$-module and $A\subset B$ a submodule of $B$ such that $p^mA=0$ form the objects in the category $S_m(R)$. We show that in case $m=2$ the categories $S_m(R)$ are in fact quite similar to each other: If also $R'$ is a commutative local uniserial ring of length $n$ and with radical factor field $k$, then the categories $S_2(R)/\mathcal N_R$ and $S_2(R')/\mathcal N_{R'}$ are equivalent for certain nilpotent categorical ideals $N_R$ and $N_{R'}$. As an application, we recover the known classification of all pairs $(B,A)$ where $B$ is a finitely generated abelian group and $A\subset B$ a subgroup of $B$ which is $p^2$-bounded for a given prime number $p$.