Topological modules over a locally compact ring

Question

Let $R$ be a locally compact, separably metrizable ring (commutative with an identity) and let $M$ be a closed submodule of $R \oplus R$. Is the projection of $M$ onto the first coordinate closed?

Answer 1

Not necessarily.

Start from $A=\mathbf{R}[t]/(t^2)$ and $R$ its closed cocompact unital subring $\{a+bt:a\in\mathbf{Z},b\in\mathbf{R}\}$. Write $J=tA$.

Then every additive subgroup of $R^2$ contained in $J^2$ is an $R$-submodule. Hence, to get a counterexample, it is enough to pick a lattice in the plane $J^2$ whose projections are not closed.