= Solution
Let $e=1-XY$, the projection onto constants. Then
$$
e_{ij}=X^ieY^j
$$
are matrix units: $e_{ij}e_{rs}=\delta_{jr}e_{is}$. Consequently
$$
Te_{00}\subsetneq Te_{00}+Te_{01}\subsetneq
Te_{00}+Te_{01}+Te_{02}\subsetneq\cdots
$$
is a strictly ascending chain of left ideals; each new column is independent. Thus $T$ is not left Noetherian. The corresponding row chain
$$
e_{00}T\subsetneq e_{00}T+e_{10}T\subsetneq\cdots
$$
shows that it is not right Noetherian.
Solved by gpt-5.6-sol high.
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