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In the book "Undergraduate Commutative Algebra" by Reid, there appears the following supposedly very easy exercise 0.5:

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Why isn't the ring $A$ of diagonal $2\times 2$ matrices over (for instance) $\mathbb{C}$, with $\alpha$ the matrix $\text{diag} \left(1,0\right)$, not a counter example? (i.e. because our $f$ is the polynomial $\left(T-1\right)T$ which is reducible.)

Thanks!

Chen Van Dam
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    Seems like there is an intended assumption that $A$ is actually a domain (or maybe that assumption is stated in the text above somewhere). – Eric Wofsey Jul 14 '21 at 06:42
  • It doesn't seem like such an assumption appears somewhere, but it does indeed solve the problem. I guess that was the intention... Thank you! – Chen Van Dam Jul 14 '21 at 07:16

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