It is known that, if $R$ is a commutative Noetherian ring, the ring of formal power series $R[[x_1,...,x_n]]$ is a flat $R$-algebra. This gives us a flat ring homomorphism $\phi:R \rightarrow R[[x_1,...,x_n]]$.
I would like to show that, if $a$ is an ideal of $R$, then $a^{ec} = a$. That is, if we take the extension of the ideal $a$, and then take the contraction of this extension, we get $a$ back again.
I've seen in certain solutions that $a^e = a + a \cdot (x_1,...,x_n)$, but I'm not sure how this is deduced. Does this simply follow from the definition of the extension of an ideal? Is $a^e$ an ideal of $R[[x_1,...,x_n]]$? My impression was that it must be to even make sense of its contraction $a^{ec}$. From there, how can we see what $a^{ec}$ would be? Do we have to study the homomorphism $\phi$ more?
Thanks!