Take $M$ an $R$-module with $R$ a commutative ring with unit. Let's take on $M$ a linear topology in the following way.
Take $M=M_0 \supset ... \supset M_n \supset M_{n+1} \supset ...$ a chain of $R$-submodules and take the topology on the group $(M,+)$ such that it's a topological group with fundamental system of neighborhoods of the neutral element is given by $M_n$.
The questions are:
If take $R$ with the discrete topology then $M$ is a topological module?
From the precedent points, I can say that for any topology on $R$ we have $M$ topological group?
If $M$ is a ring with a linear topology given like above, is $M$ a topological ring.