Ring of real-valued convergent sequences

Here is a fun and challenging problem:

Let $R$ denote the ring of real-valued convergent sequences and let $S$ denote the ring of real-valued sequences. Prove or disprove that $S\cong R$.

The cardinality of these two rings are the same (see Asaf's answer here), but I somewhat doubt the existence of a bijection that would preserve the ring structure.

I would appreciate some hints.

Source: This is the last problem in this homework sheet (as you see, the deadline is long past).

Solution 1:

Recall that a ring element $r$ is idempotent if $r^2=r$.

The ring $R$ of convergent sequences has only countably many idempotents. For a sequence $(x_n)$ is idempotent precisely if any $x_i$ is $0$ or $1$. If the idempotent sequence $(x_n)$ converges, all but finitely many of the $x_i$ are $0$, or all but finitely many are $1$.

The ring $S$ of all sequences has uncountably many idempotents.

Thus $R$ and $S$ cannot be isomorphic.

Remark: The property of having only countably many idempotents is not expressible in the first-order language of rings. It would be interesting to know whether $R$ and $S$ are elementarily equivalent.