Singular Submodules of Abelian Groups over Their Endomorphism Rings

Abstract

Let $A$ be an abelian group and $E=\Endo_{\Z}(A)$. We give a module-theoretic description of the singular $E$-submodules asked for in Fuchs' Problem~1.2. For a unital ring $R$ and a left $R$-module $M$, put $W=I_R({}_{R}R)\oplus I_R(M)$ and $J=\Jaco(\Endo_R(W))$, and let $u$ be the image of $1_R$. The classical essential-kernel criterion yields \[ Z_R(M)=M\cap Ju. \] For $R=E$ and $M=A$, all singular submodules are therefore the $E$-submodules of $A\cap Ju$. Writing $T=t(A)$ and $B=A/T$, we prove a torsion-transfer formula, identify the torsion part as ${⨁}_{p}p{T}_p$, and describe simultaneous prime lifting by a canonical obstruction. The resulting extension gives an $\Extt/\Homm$ parametrization of all singular submodules. We obtain explicit formulas for torsion groups and for $\mathbb{Z}(p^{\infty })\oplus B$ with $B$ torsion-free; in the latter case the fully invariant subgroup lattice of $B$ occurs as an interval. The general description retains the induced endomorphism action and extension data, rather than giving a classification by classical group invariants.