Nontrivial subring with identity of a ring without identity

Let K be any nontrivial unital ring.

Let $R = \left\{\left(\begin{smallmatrix} a & 0 \\ 0 & 0 \end{smallmatrix}\right) : a \in K \right\}$, and let $S = \left\{\left(\begin{smallmatrix} a & b \\ 0 & 0 \end{smallmatrix}\right) : a,b \in K \right\}$. Note that, S is a rng under the standard operations in $M_2(K)$ whereas R is a ring with identity $\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right)$.

Let $R$ be your favorite ring with $1$, let $T$ be your favorite ring without $1$, and let $S=R\times T$ (identifying $R$ with $R\times\{0\}\subset S$). Your trivial example is just the special case of this when $R$ is the trivial ring.