I am interested in the following question raised by André Nicolas in this question.
If $(R,+,\cdot, 0,1)$ is the ring of real sequences (with pointwise addition and multiplication) and $(S,+,\cdot, 0,1)$ is the subring of convergent sequences, then are $R$ and $S$ elementarily equivalent?
They are not elementarily equivalent. The ring $R$ has the (first-order) property that every non-unit is a zero-divisor. Indeed, if $x\in R$ is not a unit, then one of its coordinates is $0$. Letting $y$ be a sequence which is nonzero on that coordinate but $0$ on all other coordinates, $y\neq 0$ and $xy=0$, so $x$ is a zero-divisor.
On the other hand, $S$ does not have this property. Indeed, if $x\in S$ is any sequence of nonzero numbers which converges to $0$, then $x$ is not a unit, but it is also not a zero-divisor.
A perhaps more difficult question would be whether $S$ is elementarily equivalent to the ring of bounded sequences.