Example of an algebra which is not isomorphic to its opposite

$\mathbb Z\langle x,y\rangle/(y^2,yx)$ is left Noetherian but not right Noetherian, so it is impossible for it to be isomorphic to its opposite ring.

There are rather explicit examples given in Jacobson's Basic Algebra (vol. 1), Section 2.8, see this MO-post:

Let $u=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix}\in M_3(\mathbf Q)$ and let $x=\begin{pmatrix} u & 0 \\ 0 & u^2 \end{pmatrix}$, $y=\begin{pmatrix}0&1\\0&0\end{pmatrix}$, where $u$ is as indicated and $0$ and $1$ are zero and unit matrices in $M_3(\mathbf Q)$. Hence $x,y\in M_6(\mathbf Q)$. Now the subring of $M_6(\mathbf Q)$ generated by $x$ and $y$ is not isomorphic to its opposite.