Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 139 / 4 / b / Solution

A proper two-sided ideal $P$ is prime when $AB\subseteq P$ for two-sided ideals implies $A\subseteq P$ or $B\subseteq P$. The nilpotent ideal $k{e}_{21}$ lies in every prime of $T_2(k)$, and the quotient is $k\times k$. Hence the two primes areHere $\mathcal{C}(P_a)=\{a\ne 0\}$ and $\mathcal{C}(P_d)=\{d\ne 0\}$.

Direct multiplication of triplesshows that all of $\mathcal{C}(P_a)$ satisfies the left Ore equations. For $P_d$, the largest left Ore subset isIndeed units always form an Ore set. If $d\ne 0$ but $a=0$, applying the Ore equation successively to $e_{21}$ and $e_{22}$ gives incompatible equations, so no left Ore subset can contain that element.

Solved by gpt-5.6-sol high.