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> Assume that $A$ is a Banach algebra with two closed two sided ideals $I$ and $J$ such that $I$ and $J$ are commutative and $A=I+J$. Does this implies that $A$ is commutative? For the $C^{\*}$ algebra, the answer is ["Yes"](https://math.stackexchange.com/questions/998702/a-question-on-non-commutative-ring-or-algebra).
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**The motivation:** A (noncommutative) compact n dimensional topological manifold could be defined as follows:


A (non commutative) $C^{\*}$ or Banach algebra $A$ such that there are ideals $I\_{k}$, $k=1,2,\ldots,n$ such that $A=I\_{1}+I\_{2}+\ldots I\_{k}$ and each $I\_{j}$ is isomorphic to $C\_{0}(\mathbb{R}^{n})$. 


But the above answer in MSE shows that, in the context of $C^{\*}$ algebras, this definition does not give any non commutative example,.



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> So we search for a non commutative example in Banach algebras.
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**Note:** According to the comment on my MSE question: To what extent Banach or $C^{\*}$ algebras whose underline Lie algebras are metabelian are studied and classified?



Non commutative topological manifolds
**Theorem** Let $A$ be a unital ring and $I\_1,\dots,I\_n \subset A$ be 2-sided commutative ideals such that $A=I\_1+\dots + I\_n$. Then, $A$ is commutative.


Proof: If $A=I\_1+\dots+I\_n$, then $1 = x\_1+\dots+x\_n$ for $x\_i \in I\_i$. But then,
$$1 = (x\_1+\dots+x\_n)^{n+1} \in I\_1^2 +\dots+ I\_n^2$$ and we conclude that $A=I\_1^2 + \dots + I\_n^2$.


Now, if $I \subset A$ is any abelian 2-sided ideal, then $I^2$ is central, since
$$a(bc) = (ab)c= c(ab) = (ca)b = b(ca) = (bc)a,$$
for any $a \in A$ and $b,c \in I$.
Since $A = I\_1^2 + \dots + I\_n^2$, $A$ is commutative. q.e.d.


If $A$ is not unital, then the same argument still shows that $A^{n+1} \subset A$ is central. If $A^2 \neq A$, then there are counterexamples - even for Banach algebras. For example, one can consider the non-abelian algebra $$A = \left( \begin{matrix} 0 & \* & \* \\ 0 & 0 & \*\\0 & 0 & 0 \end{matrix}\right)$$ of strictly upper-triangular $3\times 3$-matrices (with entries in a field), which is is a sum of two abelian ideals
$$I = \left( \begin{matrix} 0 & 0 & \* \\ 0 & 0 & \*\\0 & 0 & 0 \end{matrix}\right), \quad J = \left( \begin{matrix} 0 & \* & \* \\ 0 & 0 & 0\\0 & 0 & 0 \end{matrix}\right).$$


