Baer Splitting Beyond τq-Semisimplicity
Abstract
For a commutative ring R, we call an R-module B Baer if ExtR1(B,T)=0 for torsion R-module T. It is known that every Baer module is projective when R is τq-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes τq-semisimple rings. We show that the converse fails in two substantially different ways. First, the failure is already systematic in the non-reduced Noetherian case. If D is a Dedekind domain which is not a field and R=D[ε]/(ε2), then every Baer R-module is projective, although R is not reduced, and thus is not τq-semisimple. Second, the converse fails even among reduced rings, and in fact among reduced coherent Bézout rings. The main tool is a countable-ring criterion. For a countable Noetherian domain D, let EC(D) be the ring of eventually constant sequences over D. If D is not a field, then every Baer EC(D)-module is projective, whereas EC(D) is reduced and has countably infinitely many minimal prime ideals, and thus is not τq-semisimple.
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BibTeX
@misc{zhang2026baer,
title = {{Baer Splitting Beyond $τ_q$-Semisimplicity}},
author = {Xiaolei Zhang and Guocheng Dai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14338},
eprint = {2609.14338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14338},
}
RIS
TY - UNPB
TI - Baer Splitting Beyond τ_q-Semisimplicity
AU - Xiaolei Zhang
AU - Guocheng Dai
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14338
UR - https://arxiv.org/abs/2609.14338
AB - For a commutative ring R, we call an R-module B Baer if Ext_R^1(B,T)=0 for torsion R-module T. It is known that every Baer module is projective when R is τ_q-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes τ_q-semisimple rings. We show that the converse fails in two substantially different ways. First, the failure is already systematic in the non-reduced Noetherian case. If D is a Dedekind domain which is not a field and R=D[ε]/(ε^2), then every Baer R-module is projective, although R is not reduced, and thus is not τ_q-semisimple. Second, the converse fails even among reduced rings, and in fact among reduced coherent Bézout rings. The main tool is a countable-ring criterion. For a countable Noetherian domain D, let EC(D) be the ring of eventually constant sequences over D. If D is not a field, then every Baer EC(D)-module is projective, whereas EC(D) is reduced and has countably infinitely many minimal prime ideals, and thus is not τ_q-semisimple.
ER -