The standard topology on $\mathbb{R}[[X]]$, the ring of formal power series, is the $I$-adic topology, or equivalently the product topology on $(\mathbb{R},discrete)^\mathbb{N}$. This makes $\mathbb{R}[[X]]$ into a topological ring, and makes every formal power series equal to the limit of its partial sums.
But this isn’t the only possible topology you can put on the ring of formal power series. Wikipedia says that it’s the finest topology on $\mathbb{R}[[X]]$ that makes every formal power series equal to the limit of its partial sums, but that there are coarser topologies with this same property. My question is, do these coarser topologies also make $\mathbb{R}[[x]]$ into a topological ring? If so, what are these topologies?