In Proposition 10.15 in Atiyah-Macdonlad, what does the equality $\hat{\mathfrak a}=\hat A\mathfrak a$ mean?
I know that there is an isomorphism $\hat A\otimes_A\mathfrak a\cong\hat{\mathfrak a}$ and I understand what it does (what is sent to what), but I am confused as to the 'nature' of the object $\hat A\mathfrak a$. Is it the subset of elements (Cauchy sequences) of $\hat{\mathfrak a}$ of the form $(a_ia)$ where $(a_i)$ is an element of $\hat A$ (viewed as a Cauchy sequence of elements $a_i$ of $A$) and $a\in\mathfrak a$?
If so, what does $(\hat A\mathfrak a)^n$ mean? For that matter, what sort of object is $(\hat{\mathfrak a})^n$? From what I understand $\hat{\mathfrak a}$ is a module (as all completions are), so unless it is considered as an ideal of some ring, the exponent $n$ doesn't make much sense (I am pretty sure it doesn't mean direct sum of $n$ copies).
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 $\hat{A}$.