Two-sided ideals in Leavitt path algebras

Leavitt path algebras are a natural generalization of the Leavitt algebras, which are a class of algebras introduced by Leavitt in 1962. For a directed graph $E$, the Leavitt path algebra $L_K(E)$ of $E$ with coefficients in $K$ has received much recent attention both from algebraists and analysts over the last decade, due to the fact that they have some immediate structural connections with graph $C^*$-algebras. So far, some of the algebraic properties of Leavitt path algebras have been investigated, including primitivity, simplicity and being Noetherian. We explicitly describe two-sided ideals in Leavitt path algebras associated to an arbitrary graph. Our main result is that any two-sided ideal $I$ of a Leavitt path algebra associated to an arbitrary directed graph is generated by elements of the form $(v+\sum_{i=1}^n \lambda_i g^i)(v - \sum_{e \in S} ee^*$), where $g$ is a cycle based at vertex $v$, and $S$ is a finite subset of $s^{-1}(v)$. We first use this result to describe the necessary and sufficient conditions on the arbitrary-sized graph $E$, such that the Leavitt path algebra associated to $E$ satisfies two-sided chain conditions. Then we show that this result can be used to unify and simplify many known results for Leavitt path algebras some of which have been proven by using established methodologies from $C^*$-algebras.