Let $D$ be a division ring and $S$ a subring of $D$ containing $1$. Define
$$R(D,S)=\{ (x_1, x_2, \dots,x_n, s, s, \dots)\vert n \geq 1, x_i \in D, s \in S\} $$
and say that every nonzero left (right) ideal of $R$ contains a non-zero idempotent.
I know that the ideals of $D$ are $D$ and $\{0\}$, but I don't know how to find the ideals of $R$.
Let $I$ be a non-zero left ideal of $R(D,S)$. Let $(x_{1},x_{2},\ldots,x_{n},s,s,\ldots)$ be a non-zero element in $I$. Then some entry of the vector is non-zero. If one of $x_{i}$'s is non-zero, say $x_{j}\neq 0$, then $$(0,\ldots,0,x_{j}^{-1},0,0,\ldots)(x_{1},\ldots,x_{n},s,s,\ldots)=(0,\ldots,0,1,0,0,\ldots)$$ is a non-zero idempotent in $I$, where $x_{j}^{-1}$ and $1$ are the $j$-th entry. If all $x_{i}$'s are zero, then $s\neq 0$. So $$(0,\ldots,0,s^{-1},0,0,\ldots)(x_{1},x_{2},\ldots,x_{n},s,s,\ldots)=(0,\ldots,0,1,0,0,\ldots)$$ is a non-zero idempotent in $I$, where the $s^{-1}$ and $1$ are the $(n+1)$-th entry.