Peirce Stability of Null Polynomials and a Sharp Radical Bound for Finite Rings
Abstract
Let R be a finite associative ring with identity, and let Nul(R) denote the set of polynomials in a central indeterminate that vanish identically on R under right evaluation. We establish two results concerning the failure of Nul(R) to be a right ideal. First, if e is an idempotent, f=1−e, and either eRf=0 or fRe=0, then right multiplication by e preserves Nul(R). Thus both off-diagonal Peirce components must be nonzero whenever e witnesses a failure of right stability. Second, if J is a finite nilpotent ideal whose additive group is a $2$-group and e,f are complementary idempotents such that eJf≠0, fJe≠0, J3≠0, then |J|≥32. Werner's theorem that Jac(R)3=0 implies the two-sidedness of Nul(R) therefore yields |R|≥128 whenever Nul(R) is not two-sided. We also construct a tiled matrix ring of characteristic $4$ and order $128$ whose null ideal is not two-sided.
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BibTeX
@misc{wu2026peirce,
title = {{Peirce Stability of Null Polynomials and a Sharp Radical Bound for Finite Rings}},
author = {Hongfeng Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01150},
eprint = {2609.01150},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01150},
}
RIS
TY - UNPB
TI - Peirce Stability of Null Polynomials and a Sharp Radical Bound for Finite Rings
AU - Hongfeng Wu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01150
UR - https://arxiv.org/abs/2609.01150
AB - Let R be a finite associative ring with identity, and let Nul(R) denote the set of polynomials in a central indeterminate that vanish identically on R under right evaluation. We establish two results concerning the failure of Nul(R) to be a right ideal. First, if e is an idempotent, f=1−e, and either eRf=0 or fRe=0, then right multiplication by e preserves Nul(R). Thus both off-diagonal Peirce components must be nonzero whenever e witnesses a failure of right stability. Second, if J is a finite nilpotent ideal whose additive group is a $2$-group and e,f are complementary idempotents such that eJf≠0, fJe≠0, J^3≠0, then |J|≥32. Werner's theorem that Jac(R)^3=0 implies the two-sidedness of Nul(R) therefore yields |R|≥128 whenever Nul(R) is not two-sided. We also construct a tiled matrix ring of characteristic $4$ and order $128$ whose null ideal is not two-sided.
ER -