On quasi-commutative matrices

Lengths of quasi-commutative pairs of matrices

In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the algebras generated by two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain specia...

A note on Potter’s theorem for quasi-commutative matrices

We discuss the converse of a theorem of Potter stating that if the matrix equation AB = ωBA is satisfied with ω a primitive qth root of unity, then Aq+Bq = (A+B)q. We show that both conditions have to be modified to get a converse statement and we present a characterization when the converse holds for these modified conditions and q = 3 and a conjecture for the general case. We also present som...

On quasi-zero divisor graphs of non-commutative rings

Let $R$ be an associative ring with identity. A ring $R$ is called reversible if $ab=0$, then $ba=0$ for $a,bin R$. The quasi-zero-divisor graph of $R$, denoted by $Gamma^*(R)$ is an undirected graph with all nonzero zero-divisors of $R$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exists $0neq rin R setminus (mathrm{ann}(x) cup mathrm{ann}(y))$ such tha...

Quasi-Commutative Algebras

We characterise algebras commutative with respect to a Yang-Baxter operator (quasi-commutative algebras) in terms of certain cosimplicial complexes. In some cases this characterisation allows the classification of all possible quasi-commutative structures.

A Theorem on Matrices over a Commutative Ring