Singular Submodules of Abelian Groups over Their Endomorphism Rings
Abstract
Let A be an abelian group and E=EndoZ(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=IR(RR)⊕ IR(M) and J=Jaco(EndoR(W)), and let u be the image of 1R. The classical essential-kernel criterion yields ZR(M)=M∩ Ju. For R=E and M=A, all singular submodules are therefore the E-submodules of A∩ Ju. Writing T=t(A) and B=A/T, we prove a torsion-transfer formula, identify the torsion part as ⨁p pTp, 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 Z(p∞)⊕ 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.
Related
- Representing a ring as the endomorphism ring of an abelian group
- Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs
- Invertibility of structured perturbations of singular matrices over unital rings
- Finite abelian subgroups of algebraic groups
- Hermitian K-theory for stable ∞-categories III: Grothendieck--Witt groups of rings
BibTeX
@misc{yang2026singular,
title = {{Singular Submodules of Abelian Groups over Their Endomorphism Rings}},
author = {Jiang Yang and Xin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21832},
eprint = {2609.21832},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21832},
}
RIS
TY - UNPB
TI - Singular Submodules of Abelian Groups over Their Endomorphism Rings
AU - Jiang Yang
AU - Xin Zhang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21832
UR - https://arxiv.org/abs/2609.21832
AB - 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(_RR)⊕ 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∩ Ju. For R=E and M=A, all singular submodules are therefore the E-submodules of A∩ Ju. Writing T=t(A) and B=A/T, we prove a torsion-transfer formula, identify the torsion part as ⨁_p pT_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 Z(p^∞)⊕ 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.
ER -