We follow the conventions of the previous posts regarding the ring \( \displaystyle A \).
Let \( \displaystyle \mathfrak{a} \) and \( \displaystyle \mathfrak{b} \) be two ideals of a ring \( \displaystyle A \). Prove the following:
If \( \displaystyle \mathfrak{a} \) and \( \displaystyle \mathfrak{b} \) are coprime, then \( \displaystyle \mathfrak{a} \mathfrak{b} = \mathfrak{a} \cap \mathfrak{b} \).
\( \displaystyle \mathfrak{a} \) and \( \displaystyle \mathfrak{b} \) are coprime if and only if there exist \( \displaystyle x \in \mathfrak{a} \) and \( \displaystyle y \in \mathfrak{b} \) such that \( \displaystyle x + y = 1_{A} \).
Can you generalise the second result (that is, find a characterisation of "coprime") for a finite family \( \displaystyle \{ \mathfrak{a}_{i} \} _{i=1}^{n} \) of ideals of \( A \)?
On the other hand, if \(\displaystyle{r\in a\,b}\) , then \(\displaystyle{r=\sum_{i=1}^{n}x_{i}\,y_{i}}\) for some \(\displaystyle{n\in\mathbb{N}}\), where :