TITLE: A ring for which the category of left and right modules are distinct
QUESTION [7 upvotes]: What is an example of a ring $R$ for which the abelian category of left $R$-modules is not isomorphic to the category of right $R$-modules?

REPLY [19 votes]: $\begin{pmatrix} {\mathbb Z} & {\mathbb Q}\\ 0 & {\mathbb Q} \end{pmatrix}$
is a canonical example for such things. Let us list all simple right and left modules:
$$R_p=\begin{pmatrix} {\mathbb Z}/(p) & 0\end{pmatrix}, \ 
R_0 =\begin{pmatrix} 0 & {\mathbb Q} \end{pmatrix}, \ 
L_p=\begin{pmatrix} {\mathbb Z}/(p) \\ 0 \end{pmatrix}, \ 
L_0=\begin{pmatrix} 0 \\ {\mathbb Q} \end{pmatrix}$$
for various primes $p$. Now notice that $R_0$ is a projective simple right module. By inspection, there exists no projective simple left module.