Two-Sided Noncommutative Gröbner Basis on Quiver Algebras

For a quiver $Q$, we define a path algebra $KQ$ as a span of all the paths of positive length. We study left (respective right) sided ideals and their Gröbner bases. We introduce the two-sided ideals, two-sided division algorithm for elements of $KQ$ and study the two-sided Gröbner bases. We show that with the defined two-sided division algorithm and two-sided Buchberger’s algorithm, we can find a finite or an infinite Gröbner basis for a two-sided ideal $I\subseteq KQ$ given a fixed admissible ordering.