Matrix rings and zero divisors

Ring. $M_2(\mathbb{R})$ with usual addition and multiplication is a non-commutative ring with unit $I_2$.

Zero divisors. $\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\1&0\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix}$, with both factors nonzero.

Conclusion. $M_2(\mathbb{R})$ is a non-commutative ring with unit, not an integral domain.