I'm self-learning commutative algebra following "Introduction to Commutative Algebra". When dealing with concepts like "contraction" and "extension", some exercises in this book don't specify which homomorphism it uses and make it hard to understand, like this problem:
Let $A$ be a ring and let $A[[x]]$ be the ring of formal power series with coefficients in $A$. Show that the contraction of a maximal ideal $m$ of $A[[x]]$ is maximal ideal of $A$, and $m$ is generated by $m^c$ and $x$. (Here $m^c$ is the contraction of $m$.)
In this problem, the author doesn't specify the homomorphism he uses. Is it the inclusion mapping $f(a) = a$??? And please help me to solve this problem, too. Thanks. I really appreciate.