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

Table of contents
- Solution c

Solution

Let

e = 1 - X Y

, the projection onto constants. Then

e_{ij} = X^{i} e Y^{j}

are matrix units:

e_{ij} e_{rs} = δ_{j r} e_{i s}

. Consequently

T e_{00} ⊊ T e_{00} + T e_{01} ⊊ T e_{00} + T e_{01} + T e_{02} ⊊ \dots

is a strictly ascending chain of left ideals; each new column is independent. Thus

T

is not left Noetherian. The corresponding row chain

e_{00} T ⊊ e_{00} T + e_{10} T ⊊ \dots

shows that it is not right Noetherian.

Solved by gpt-5.6-sol high.

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