I don't understand the symbol $\widehat A \mathfrak{a}$. I saw the following post, but I still have the question.
Atiyah-Macdonald, Chapter 10, Proposition 10.15 clarifications There is following answer.
Suppose $R\rightarrow S$ is a ring homomorphism and $\mathfrak{a}$ is an ideal of $R$. Do you know what $\mathfrak{a}S$ means? It means the ideal in $S$ generated by the image of $\mathfrak{a}$ under the homomorphism. $\hat{A}\mathfrak{a}$ means the same thing here. And yes, $\hat{\mathfrak{a}}$ is an ideal of $\hat{A}$.
From the answer of the post, I think $\widehat{A}\mathfrak{a}$ is the following meaning.
Let $f_A:A\to \widehat{A}$ be natural homomorphism. Then $\widehat{A}\mathfrak{a}=\widehat{A}f_A(\mathfrak{a})$
But,$f_A$ is a ring homomorphism that is not written on the proof but my idea.
Followings is my idea.
Let $f_\mathfrak{a}:\mathfrak{a}\to \widehat{\mathfrak{a}}$ be natural homomorphism. Then I think $\phi:\hat{A}\otimes \mathfrak{a}\to \hat{\mathfrak{a}}$ is $\sum a_i\otimes x_i\mapsto \sum a_if_\mathfrak{a}(x_i)$. Therefore the image of $\phi$ is $\widehat{A}f_\mathfrak{a}(\mathfrak{a})$.
But it is not natural to write $\widehat{A}f_\mathfrak{a}(\mathfrak{a})$ as $\widehat{A}\mathfrak{a}$.
So I'm wondering if the author meant something else by this symbol $\widehat{A}\mathfrak{a}$.
Please tell me the correct meaning.
