Two-Sided Noncommutative Gr\"{o}bner Basis on Quiver Algebras

Abstract

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 Gr\"{o}bner bases. We introduce the two-sided ideals, two-sided division algorithm for elements of $KQ$ and study the two-sided Gr\"{o}bner 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 Gr\"{o}bner basis for a two-sided ideal $I \subseteq KQ$ given a fixed admissible ordering.