In Bourbaki's Algebra there is the following proposition:
Let $A$ be a ring (with $1$), $(x_\lambda)_{\lambda\in L}$ a family of elements of $A$ and $\mathfrak{a}$ the set of sums $\sum_{\lambda\in L}a_\lambda x_\lambda b_\lambda$ where $(a_\lambda)_{\lambda\in L}$, $(b_\lambda)_{\lambda\in L}$ are families with finite support of elements of $A$. Then $\mathfrak{a}$ is the two-sided ideal of $A$ generated by the elements $x_\lambda$.
The proof, they say, is analogous to the corresponding statement for left ideals.
However, I wonder if that is true. Let's consider the case $|L|=1$. I don't see at all, how one can, for $x,a,a',b,b'\in A$, find $\alpha, \beta\in A$ such that $axb+a'xb'=\alpha x\beta.$
The form of the definition of two-sided principal ideal on Wikipedia strengthens my doubts.
Can somebody clear this up?