# Left-ideals in subalgebras

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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The answer is no. Let $A = \mathbb{C}(x)$ and $B = \mathbb{C}[x] \subset A$, i.e. the field of rational functions in one variable and the subring of polynomials. Consider $\mathcal{L} = 0$, the zero ideal in $A$.
If $A$ and $B$ are commutative, finitely generated, and have the same unity element, the answer is yes: this is nontrivial and closely related to the Nullstellensatz. This counterexample shows that the finitely generated hypothesis is necessary, but I am unsure about the rest.