Leavitt path algebras are Bézout

Let $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. We show that $L_K(E)$ is a Bézout ring, i.e., that every finitely generated one-sided ideal of $L_K(E)$ is principal.

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