What are the simplest examples of
rings that are not isomorphic to their
opposite rings? Is there a science to constructing them?




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The only simple example known to me:
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In Jacobson's Basic Algebra (vol. 1), Section 2.8, there is an exercise that goes as follows:


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 $x,y\in M\_6(\mathbf Q)$. Jacobson gives hints to prove that the subring of $M\_6(\mathbf Q)$ generated by $x$ and $y$ is not isomorphic to its opposite.


Examples seem to be well-known to the operator algebras crowd:
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See for example the paper: "A Simple Separable C\*-Algebra Not Isomorphic to Its Opposite Algebra" by N. Christopher Phillips, Proceedings of the American Mathematical Society
Vol. 132, No. 10 (Oct., 2004), pp. 2997-3005.



Simplest examples of rings that are not isomorphic to their opposites
Here's a factory for making examples. If $\Gamma$ is a quiver, and $k$ a field, then we get a quiver algebra $k\Gamma$. If $\Gamma$ has no oriented cycles, we can recover $\Gamma$ from $k\Gamma$ by taking the Ext-construction. Also, the opposite algebra of a quiver algebra is obtained by reversing all the arrows in the quiver.


Hence you can produce an example by taking the quiver algebra of any quiver with no oriented cycles, which is not isomorphic to its reverse. It's easy to construct lots of quivers with these properties.


