It the corollary $7.17$ in the commutative algebra book of David Eisenbud, as follows:
I could not understand the highlighted line.
What does mean by the line $\text{image of $\varphi(x)=f$ of the generator $X$ of ${\color{blue}{(x)}}$ of ${\color{red}{(x)}}$}$ ?
I understand that the ideal ${\color{blue}{(x)}}$ from the domain ring $R[[x]]$ and the ideal ${\color{red}{(x)}}$ is from the co-domain ring $R[[x]]$.
But why $X$ is used here instead of $x$ ?
The image of the generator $X$ should be $\varphi(X)=f$ instead of $\varphi(x)=f$.
Can you explain please ?
