A note on the sum of two closed lattice ideals

Suppose that $E$ is a locally convex lattice. The main results established in this note are: (a) If $I,J$ are $\sigma (E',E)$-closed lattice ideals in the dual $E'$ of $E$, then $I + J$ is $\sigma (E',E)$-closed. (b) If $E$ is a Fréchet lattice (in particular, if $E$ is a Banach lattice) and if $I,J$ are closed lattice ideals in $E$, then $I + J$ is closed.