Semicommutativity via Zero-Insertive Elements: Insights from Path and Leavitt Path Algebras

In the path algebra $KE$, where $K$ is a field and $E$ is a directed graph (or quiver), every non-loop edge can be expressed in the form $arb$, where $a,b,r\in KE$ and $ab=0$. This fact leads us to the characterization that $KE$ is semicommutative if and only if $E$ contains no non-loop edges. For a