A question on non commutative ring or algebra

Question

Assume that $R$ is a ring such that $R=I+J$ where $I$ and $J$ are 2 -sided ideal.(This is not a direct sum) If $I$ and $J$ are commutative does it implies that $R$ is a commutative ring? Please consider the same question for $C^{*}$ algebra $A$. (In this question $R$ or $A$ can be unital or non unital) If the answer is no, what is the counter example in the area of $C^{*}$ algebras?

Answer 1

A closed commutative ideal in a C$^*$-algebra is necessarily contained in the centre of the algebra (proof below). So if $A=I+J$ with $I,J$ commutative ideals, then $A$ is commutative as $A\subset Z(A)\subset A$, so $A=Z(A)$.

To show that a commutative ideal is central: let $x\in A$ and $y\in J^+$, where $J$ is a commutative ideal. Then $y=z^*z$, with $z\in J$ (as $J$ is a C$^*$-algebra). Then $$ xy=xz^*z=(xz^*)z=z(xz^*)=(zx)z^*=z^*zx=yx. $$ As the positive elements in $J$ span $J$, we get $xy=yx$ for all $x\in A$, $y\in J$.