Give an example of an injective ring homomorphism $f : R \to S$ where $R$ is commutative, but $S$ is not commutative

Question

I feel like matrices would be the right way to go for something that doesn't commute. However, the matrix

$$ \begin{pmatrix} a & b \\ b & a \end{pmatrix}$$

commutes.

Answer 1

Consider any diagonal embedding $R\longrightarrow M_n(R)$ for $n>1$, i.e. send $r\mapsto {\rm diag}(r,\ldots,r)$. A simpler example is $Z(R)\hookrightarrow R$ for any noncommutative ring $R$!

Answer 2

You should be aware of the fact that your homomorphism must not be surjective. You're probably thinking to $$ \mathbb{C}\to M_2(\mathbb{R}), \qquad a+bi\mapsto\begin{bmatrix}a & b\\ -b &a\end{bmatrix} $$ which is a good example.

Easier, the map $$ F\to M_2(F), \qquad a\mapsto\begin{bmatrix}a & 0\\0 &a\end{bmatrix} $$ where $F$ is a field and $M_2(F)$ is the ring of $2\times2$ matrices.