Let $R$ be a commutative unital ring and $S$ an associative rng (nonunital) that is an $R$-module (= $(R,R)$-bimodule with $\forall s \in S, r \in R$: $rs = sr$). The Dorroh extension is a well-known construction that "adjoins unity to $S$ using $R$", i.e. constructs a unital ring $T$ with $S \lhd T$ and $R \subset T$. The additive group of $T$ is $R \oplus S$, and the multiplication is defined as $(r,s)(r',s') = (rr',rs'+sr'+ss')$. This looks like a "bilinear form".
Question: is there a "matrix construction" of $T$, like that of trivial extensions: $\{\begin{pmatrix} r & s \\ & r\end{pmatrix}: r \in R, s \in S\}$, and triangular rings: $\{\begin{pmatrix} r & m \\ & s\end{pmatrix}: r \in R, m \in M, s \in S\}$? I.e. is the ring $T$ isomorphic to $\{(a_{ij})\}$, where $a_{ij} \in R$ whenever $(i,j) \in M$ and $a_{ij} \in S$ if $(i,j)$ is in the complement of $M$ in $\{1,\ldots,n\}^2$, with some (perhaps linear) relations pre-imposed on $a_{ij}$'s?
With $\phi:(a,b)\mapsto \begin{bmatrix}a&0\\n&a+n\end{bmatrix}$
You have an injective additive map that also satisfies
$$ \phi((a,n))\phi((b,m))=\begin{bmatrix}a&0\\n&a+n\end{bmatrix}\begin{bmatrix}b&0\\m&b+m\end{bmatrix}=\\\begin{bmatrix}ab&0\\nb+am+nm&ab+am+nb+nm\end{bmatrix}=\\\begin{bmatrix}ab&0\\am+nb+nm&ab+(am+nb+nm)\end{bmatrix}=\phi((a,n)(b,m)) $$
It's perhaps not as tidy as the other two, but it's not far off.