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

Let K be a field and LK(E,w) the Leavitt path algebra of a row-finite weighted graph $(E,w)$. Suppose that LK(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 LK(E,w). The embedding preserves multiplication of ideals modulo the lower endpoint of the interval. It follows that the ideal lattice of LK(E,w) is not distributive and that ideal multiplication in LK(E,w) is not commutative. Moreover, we show that if $(E,w)$ is countable, then LK(E,w) has exactly |K|₀ ideals, prime ideals, left primitive ideals, and right primitive ideals.