Let $A$ be a Banach algebra, $I$ be a closed two-sided ideal in $A$, and $J$ be a closed two-sided ideal in $I$ such that there is no ideal between $I$ and $J$. Can we see $dim(\frac{I}{J})<\infty$?

Y. DOMAR in "On the ideal structure of certain Banach algebras", proves a lemma like this question with the following difference:

$A$ is commutative, $J$ is a two-sided ideal in $A$ and he shows $dim(\frac{I}{J})=1$