A proper two-sided ideal $P$ is <prime ideal of a noncommutative ring>[prime] when $AB\subseteq P$ for two-sided ideals implies $A\subseteq P$ or $B\subseteq P$. The nilpotent ideal $ke_{21}$ lies in every prime of $T_2(k)$, and the quotient is $k\times k$. Hence the two primes are
$$
P_a=\{(a,c,d):a=0\},\qquad
P_d=\{(a,c,d):d=0\}.
$$
Here $\mathcal C(P_a)=\{a\ne0\}$ and $\mathcal C(P_d)=\{d\ne0\}$.

Direct multiplication of triples
$$
(a,c,d)(a',c',d')=(aa',ca'+dc',dd')
$$
shows that all of $\mathcal C(P_a)$ satisfies the left Ore equations. For $P_d$, the largest left Ore subset is
$$
\{(a,c,d):a\ne0,\ d\ne0\}=T_2(k)^*.
$$
Indeed units always form an Ore set. If $d\ne0$ 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.

Solution

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

{wiki=Prime_ideal#Prime_ideals_for_noncommutative_rings}

A proper two-sided ideal $P$ of a noncommutative ring is prime when $AB\subseteq P$ for two-sided ideals $A,B$ implies $A\subseteq P$ or $B\subseteq P$.


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