Atiyah Macdonald Exercise 1.5


I'm trying to prove that:

The contraction of a maximal ideal $ mathfrak {m} $ of $ A[[x]]]$ is a maximal ideal of $ A $, and $ mathfrak {m} $ is generated by $ mathfrak {m} ^ c $ and $ x $.

I have made the following progress:

assume $ x not in mathfrak {m} $. then $ 1-xg in mathfrak {m} $ for some people $ g in A[[x]]$, and this is a contradiction because $ 1-xg is a unit. So, $ x in mathfrak {m} $. Consider any non-zero element $ a in A $. We want to show that there is $ b in A $ such as $ ab equiv 1 ( mathfrak {m} ^ c) $. Since $ A[[x]]/ mathfrak {m} $ is a field we know that there is $ f = sum a_nx ^ n in A[[x]]$ such as $ af equiv 1 ( mathfrak {m}) $, and since $ x in mathfrak {m} $, $ aa_0 equiv 1 ( mathfrak {m}) $. How can I conclude from this that $ aa_0 equiv 1 ( mathfrak {m} ^ c) $ ?

Thank you.