For each $ n \in \mathbb{N} $, the matrix algebra $ {\mathbb{M}_{n}}(\mathbb{C}) $ has indeed no non-trivial proper ideals. A proof of this is as follows.
Proof: For notational convenience, let $ [n] = \{ 1,\ldots,n \} $. For each $ (i,j) \in [n] \times [n] $, let $ I_{i j} $ denote the $ (n \times n) $-matrix with $ (i,j) $-th entry $ 1 $ and $ 0 $’s elsewhere. Then observe that
$$
\forall (i,j),(\alpha,\beta) \in [n] \times [n]: \quad
I_{i j} I_{\alpha \beta} = \delta_{j \alpha} I_{i \beta},
$$
where $ \delta $ is Kronecker’s delta. For the sake of contradiction, suppose that there exists a non-trivial proper ideal $ J $ of $ {\mathbb{M}_{n}}(\mathbb{C}) $. Pick a non-zero element $ A \in J $, and write $ A $ as a linear combination of the $ I_{i j} $’s:
$$
A = \sum_{i,j = 1}^{n} a_{i j} I_{i j}, \quad \text{where $ a_{i j} \in \mathbb{C} $.}
$$
Then $ a_{r s} \neq 0 $ for some $ (r,s) \in [n] \times [n] $. As $ I_{r s} A I_{s r} = a_{r s} I_{r r} \in J $ (check!), we have $ I_{r r} \in J $. Hence, $ I_{i j} = I_{i r} I_{r r} I_{r j} \in J $ for all $ (i,j) \in [n] \times [n] $, which yields $ J = {\mathbb{M}_{n}}(\mathbb{C}) $. Contradiction. $ \quad \spadesuit $
There are many other examples of Banach algebras with no non-trivial closed proper ideals. Here are a few well-known $ C^{*} $-algebraic examples (we call them simple $ C^{*} $-algebras).
The algebra of compact linear operators on a Hilbert space $ \mathcal{H} $, denoted by $ \mathcal{K}(\mathcal{H}) $. (This is a standard example and is mentioned by Branimir Ćaćić in his comment above. It subsumes the example of $ {\mathbb{M}_{n}}(\mathbb{C}) $, since $ {\mathbb{M}_{n}}(\mathbb{C}) \cong \mathcal{B}(\mathbb{C}^{n}) = \mathcal{K}(\mathbb{C}^{n}) $.)
For each irrational $ \theta \in \mathbb{R} $, the irrational rotation algebra $ A_{\theta} $. This is the universal $ C^{*} $-algebra generated by unitaries $ u $ and $ v $ subject to the relation $ u v = e^{2 \pi i \theta} v u $. (To learn more about $ A_{\theta} $, please consult the paper $ C^{*} $-Algebras Associated with Irrational Rotations by Marc Rieffel.)
For each $ n \in \mathbb{N}_{\geq 2} $, the Cuntz algebra $ \mathcal{O}_{n} $. This is the universal $ C^{*} $-algebra generated by $ n $ isometries $ S_{1},\ldots,S_{n} $ subject to the relation $ \displaystyle \sum_{i = 1}^{n} S_{i} S_{i}^{*} = 1 $. (To learn more about $ \mathcal{O}_{n} $, please consult the paper Simple $ C^{*} $-Algebras Generated by Isometries by Joachim Cuntz.)
The reduced group $ C^{*} $-algebra $ {C^{*}_{r}}(G * H) $, where $ G $ and $ H $ are discrete groups not both of order $ 2 $. (This example is due to William Paschke and Norberto Salinas, and it is the subject of their paper $ C^{*} $-Algebras Associated with Free Products of Groups.)
Once you know some simple $ C^{*} $-algebras, you can use the following result to create others.
Theorem (Takesaki): If $ A $ and $ B $ are simple $ C^{*} $-algebras, then so is $ A \otimes_{\min} B $.
Unfortunately, I cannot think of non-$ C^{*} $-algebraic examples, but I hope that this answer suffices. Good luck with your search!
First, we need to show that your homomorphism is surjective. Since these two ideals are relatively prime, you can find $e_{1} \in I$ and $e_{2} \in J$ such that $e_{1}+e_{2}=1$ $f(e_{1})=(e_{1}+I,e_{2}+J)=(e_{1}+I,1-e_{2}+J)=(\overline{0},\overline{1})$. You can do the similar thing to show $f(e_{2})=(\overline{1},\overline{0})$. Can you deduce surjectivity then? The kernel is $I \cap J$. We need to show $I \cap J =IJ$. Notice that $IJ \subset I \cap J$ is clear. Now let $x \in I \cap J$. Write $x=x \times 1=x(e_{1}+e_{2})=xe_{1}+xe_{2}$. Could you see what to do next ?
Best Answer
Here are two facts that justify saying that "normal subgroups are to groups" as "ideals are to rings", but that's just about all we can say.
(1) Normal subgroups are precisely the kernels of group morphisms.
(1') Ideals are precisely the kernels of ring morphisms.
(2) Normal subgroups are the only subsets on which the natural group structure on the cosets is well-defined.
(2') Ideals are the only subsets on which the naturall ring structure on the cosets is well-defined.
In short, both are the only objects that allow us to look at quotient structures.