Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 139 / 1 / c / Solution

Let $e=1-XY$, the projection onto constants. Thenare matrix units: $e_{ij}{e}_{rs}={\delta }_{jr}{e}_{is}$. Consequentlyis a strictly ascending chain of left ideals; each new column is independent. Thus $T$ is not left Noetherian. The corresponding row chainshows that it is not right Noetherian.