Ideal in Matrix algebra

Question

Let A be a Banach algebra and suppose that $\Lambda$ be a nonempty set. With the following product and norm the matrix space $$M_\Lambda(A)=\{(a_{ij})_{i,j\in\Lambda}:\sum_{i,j\in\Lambda}\|a_{ij}\|<\infty,\quad a_{ij}\in A\}$$ is Banach algebra: $$(a_{ij})_{i,j\in\Lambda}.(b_{ij})_{i,j\in\Lambda}=(c_{ij})_{i,j\in\Lambda}$$ where $c_{ij}=\sum_{k\in\Lambda}a_{ik}b_{kj}$, and $$\|(a_{ij})_{i,j\in\Lambda}\|=\sum_{i,j\in\Lambda}\|a_{ij}\|$$ Now if $I$ be an ideal in $M_\Lambda(A)$, could we say that there exists an ideal $I_0$ in $A$ such that $I=M_\Lambda(I_0)$? If it does not exists in general when it exists?How about in abstract algebras with finite set $\Lambda$?