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arXiv · 2611.07521

On the Connected Components of Structure Graphs of Associative Rings

Andrey M. Chernev·Nikolay A. Chernev
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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 $\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_7$, $GF(4)$, and their direct sums. The hypothesis is proved for all rings except residue rings modulo a prime of the form $7+12k$, $k\geq181$, and Galois fields of characteristic~2 with extension degree $\geq17$.

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Andrey M. Chernev, Nikolay A. Chernev. 2026-11-05. On the Connected Components of Structure Graphs of Associative Rings. https://arxiv.org/abs/2611.07521

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