Small's example of left hereditary ring but not right hereditary Let $R=\left( \begin{array}{ccc} \mathbb{Z}&0\\ \mathbb{Q}&\mathbb{Q} \end{array}\right)$. 
It is known to be left hereditary but not right hereditary. But I don't know how to prove it. Please give me some hints or answers.
 A: A correction: this ring is left hereditary and not right hereditary.
It has been covered here how to show that the strictly triangular subring is a one-sided ideal that isn't projective, which would prove that it is not right hereditary (after you adapt it for the side change.)
A proof appears on page 46 of Lectures on rings and modules by T.Y. Lam. Using the opposite ring instead of your version.  The strategy used is to use the classification of right ideals (left ideals in your case) to show that they are all projective.
It turns out that there are three types of left ideals:
$A_n=\begin{bmatrix}n\mathbb Z&0 \\ \mathbb Q&\mathbb Q\end{bmatrix}$
$B_n=\begin{bmatrix}n\mathbb Z&0 \\ \mathbb Q&0\end{bmatrix}$
$C_V=\left\{\begin{bmatrix}0&0 \\ p&q\end{bmatrix}\middle |(p,q)\in V, V \text{ a subspace of $\mathbb Q^2$}\right\}$
The types $A_n$, $n\neq 0$ are all free, and because they split into $A_n\cong B_n\oplus\begin{bmatrix}0&0\\ 0& \mathbb Q\end{bmatrix}$, the $B_n$ are also projective.
The trickiest part is to show the ones of the form $C_V$ are projective, and I don't have enough time at the moment to give a complete argument, but I will when I can.   Until then I hope the reference and the clue will get you on your way.
