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On the Connected Components of Structure Graphs of Associative Rings
Abstract
The article studies the notion of the structure graph of an associative ring with unity. It is known that the structural graph of a ring with n elements has at least n connected components. The article investigates when this lower bound is attained. A hypothesis is stated that the only rings of order n having exactly n connected components are the rings ℤ2, ℤ3, ℤ7, GF(4), and their direct sums. The hypothesis is proved for all rings except residue rings modulo a prime of the form 7+12k, k≥181, and Galois fields of characteristic~2 with extension degree ≥17.
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BibTeX
@misc{chernev2026connected,
title = {{On the Connected Components of Structure Graphs of Associative Rings}},
author = {Andrey M. Chernev and Nikolay A. Chernev},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07521},
eprint = {2611.07521},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07521},
}
RIS
TY - UNPB
TI - On the Connected Components of Structure Graphs of Associative Rings
AU - Andrey M. Chernev
AU - Nikolay A. Chernev
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07521
UR - https://arxiv.org/abs/2611.07521
AB - The article studies the notion of the structure graph of an associative ring with unity. It is known that the structural graph of a ring with n elements has at least n connected components. The article investigates when this lower bound is attained. A hypothesis is stated that the only rings of order n having exactly n connected components are the rings ℤ_2, ℤ_3, ℤ_7, GF(4), and their direct sums. The hypothesis is proved for all rings except residue rings modulo a prime of the form 7+12k, k≥181, and Galois fields of characteristic~2 with extension degree ≥17.
ER -