Suppose $\mathfrak{A}$ is an unital algebra over complex numbers and $\mathfrak{J}$ is chain of left-ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably generated. Clearly, the union $\bigcup \mathfrak{J}$ is a left-ideal. Can it be countably generated? I am interested in the commutative case as well.

Recently, I asked a similar question for Boolean algebras but I prefer these two questions not to be merged.