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Suppose $A$ and $B$ are (complex) unital associative algebras (with not necessarily the same units) and $B\subseteq A$. Also, let $\mathcal{L}$ be a maximal left ideal in $A$. Is it true that either $B\subseteq\mathcal{L}$ or $B\cap \mathcal{L}$ is a maximal left ideal in $B$?

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