I am looking for an example in noncommutative ring theory. Namely, I am looking for a left perfect ring with finite left global dimension that is not right coherent. It seems to me that should be possible to get such example, but did not find it yet...
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It seems the ring $$A=\begin{bmatrix} \mathbb Q & \mathbb Q & \mathbb R\\ 0 & \mathbb Q & \mathbb R\\ 0 & 0 & \mathbb Q \end{bmatrix} / \begin{bmatrix} 0 & 0 & \mathbb R\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}$$
is such an example. It is semiprimary, hence perfect on both sides, and the global dimension is $2$.
The right ideal $$I=\begin{bmatrix} 0 & \mathbb Q & \mathbb R\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} / \begin{bmatrix} 0 & 0 & \mathbb R\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} $$ is finitely generated, but not finitely presented, so $A$ is not right coherent.
answered Mar 15, 2015 at 23:23