In $R=\mathbb Z[\sqrt{10}]=\{a+b\sqrt{10}:a,b\in\mathbb Z\}$, let I and J be ideals generated by $\{5,\sqrt{10}\}$ and $\{4,\sqrt{10}\}$ respectively. Which statement holds?
Correct answer: (B) I is maximal but J is not prime.
Explanation
For the printed generators, $R/I\cong\mathbb F_5$, so I is maximal. Also J contains $(\sqrt{10})^2-2(4)=2$, hence $J=(2,\sqrt{10})$ and $R/J\cong\mathbb F_2$. Thus J is maximal and prime as well, making A the mathematical answer to the printed question. The appended official key gives B, which conflicts with these quotient calculations. Historical grading B is retained with this explicit source/key conflict; the explanation does not claim that J is nonprime.
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