arXiv subjects
Publications and source records attributed to C. Miguel.
We consider a generalization of the quaternion ring $\Big(\frac{a,b}{R}\Big)$ over a commutative unital ring $R$ that includes the case when $a$ and $b$ are not units of $R$. In this paper, we focus on the case $R=\mathbb{Z}/n\mathbb{Z}$ for and odd $n$. In particular, for every odd integer $n$ we compute the number of non-isomorphic generalized quaternion rings $\Big(\frac{a,b}{\mathbb{Z}/n\mathbb{Z}}\Big)$
For a division ring D, finite dimensional over its center F, we give a condiction for the connectedness of the commuting graph of a matrix ring over $D$. Furthermore, we prove that if the commuting graph is connected, then its diameter is between four and six.
We will give a short proof of the fact that if the algebraic closure of a field $\mathbb F$ is a finite extension, then for $n\geq 3$ the commuting graph $Γ(M_n(\mathbb F))$ is connected and its diameter is four.
We prove that the diameter of the commuting graph of the full ma- trix ring over the real numbers is at most five. This answers, in the affir- mative, a conjecture proposed by Akbari-Mohammadian-radjavi-Raja, for the special case of the field of real numbers.