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 $\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_7$, $GF(4)$, 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 $\geq17$.