A note on the ideal lattice of a weighted Leavitt path algebra

Let $K$ be a field and $L_K(E,w)$ the Leavitt path algebra of a row-finite weighted graph $(E,w)$. Suppose that $L_K(E,w)$ is not isomorphic to an ordinary Leavitt path algebra. We prove that the ideal lattice of every countably generated unital $K$-algebra embeds as a closed interval into the ideal lattice of $L_K(E,w)$. The embedding preserves multiplication of ideals modulo the lower endpoint of the interval. It follows that the ideal lattice of $L_K(E,w)$ is not distributive and that ideal multiplication in $L_K(E,w)$ is not commutative. Moreover, we show that if $(E,w)$ is countable, then $L_K(E,w)$ has exactly $|K|^{\aleph_0}$ ideals, prime ideals, left primitive ideals, and right primitive ideals.