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Let $A$ and $B$ be complex unital algebras and $h\colon A\to B$ be an injective (unital) algebra homomorphism. Let $\mathcal{L}$ be a left-ideal in $A$ such that $B\cdot h(\mathcal{L})$ (left-ideal generated by $h(\mathcal{L})$) is a singly generated maximal left-ideal in $B$. Is $\mathcal{L}$ a maximal left-ideal in $A$?

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Let $A=C[x]$, $B=C(x)$. Let $h:A\to B$ be the inclusion. Let $I=(0)\subseteq A$. Then $B h(I)$ is a principle left ideal which is maximal, but $I$ is not a maximal ideal in $A$.

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