  We explicitly describe two-sided ideals in Leavitt path algebras associated
with a row-finite graph. Our main result is that any two-sided ideal $I$ of a
Leavitt path algebra associated with a row-finite graph is generated by
elements of the form $v + \sum_{i=1}^n\lambda_i g^i$, where $g$ is a cycle
based at vertex $v$. We use this result to show that a Leavitt path algebra is
two-sided Noetherian if and only if the ascending chain condition holds for
hereditary and saturated closures of the subsets of the vertices of the
row-finite graph $E$.
