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Let $A$ be a complex unital algebra and let $M$ be a proper left ideal in $A$. Furthermore, suppose that $\{L_i\colon i \in I\}$ is an uncountable family of left-ideals such that

1) for each $i$ the $L_i\subseteq M$ and $\mbox{dim }M/L_i = 2$

2) for $s\neq t$ neither $L_s\subseteq L_t$ nor $L_t\subseteq L_s$.

Can $M$ be finitely generated?

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