Embedding into quotient
Abstract
We study Bass's finiteness problem for embeddings into proper quotients. Given a commutative ring R, a reduced finitely generated R-algebra A, and an ideal Jtriangleleft A, we ask whether an R-algebra monomorphism A↪R A/J forces J=0. We give affirmative answers by presenting two arguments. Also, we discuss not only in the reduced case, but also in several non-reduced cases: when R is artinian and A is finite over R; when R is zero-dimensional and A is finite projective over R; and in a series of Cohen--Macaulay settings. In the module-theoretic direction, for a noetherian reduced ring, and a finitely generated, and torsion-free A-module M, we show that M↪R M/JM implies that J is contained in a minimal prime of A; if M is faithful, then J=0. Also, dual versions investigated. Examples illustrate the necessity of hypotheses.
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BibTeX
@misc{asgharzadeh2026embedding,
title = {{Embedding into quotient}},
author = {Mohsen Asgharzadeh and Elham Mahdavi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25845},
eprint = {2610.25845},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25845},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Embedding into quotient
AU - Mohsen Asgharzadeh
AU - Elham Mahdavi
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25845
UR - https://arxiv.org/abs/2610.25845
AB - We study Bass's finiteness problem for embeddings into proper quotients. Given a commutative ring R, a reduced finitely generated R-algebra A, and an ideal Jtriangleleft A, we ask whether an R-algebra monomorphism A↪_R A/J forces J=0. We give affirmative answers by presenting two arguments. Also, we discuss not only in the reduced case, but also in several non-reduced cases: when R is artinian and A is finite over R; when R is zero-dimensional and A is finite projective over R; and in a series of Cohen--Macaulay settings. In the module-theoretic direction, for a noetherian reduced ring, and a finitely generated, and torsion-free A-module M, we show that M↪_R M/JM implies that J is contained in a minimal prime of A; if M is faithful, then J=0. Also, dual versions investigated. Examples illustrate the necessity of hypotheses.
ER -