How can the only maximal ideal of $C[x] / X^2$ be $(X)$?

Question

In my notes I have the following example which I don't understand.

Let $f$ be the canonical injection from $C$ to $C[X]/X^2$.The only maximal ideal of $C[X]/X^2$ is $(X)$ and $f^{-1}((X))$=$(0)$.

What I don't understand are the following what does $f$ do? It takes an element $p$ in $C$ and assigns it to $p+(X^2)$? And how can the maximal ideal of $C[X]/X^2$ be $(X)$? Elements of the former are cosets and elements of the latter are polynomials.

Thank you in advance.

Answer 1

The only maximal ideal of $C[X]/X^2$ is the image of the ideal $(X) \subset C[X]$ in $C[X]/X^2.$ It is normally written as the ideal generated by $X + (X^2)$ or simply $(\bar X)$ where $\bar X$ denotes the image of $X$ in $C[X]/X^2$ (under the canonical projection). Sometimes people just write $(X)$ in $C[X]/X^2$ if there is no confusion. But one should keep in mind that it means "the image of the ideal $(X) \subset C[X]$ in $C[X]/X^2.$"

Answer 2

(1) You have the definition of $f$ right. You could think of it as the composition $$ \mathbf{C} \to \mathbf{C}[X] \to \mathbf{C}[X]/(X^2) $$ where the first map is the inclusion of $\mathbf{C}$ as scalar polynomials and the second is the quotient map.

(2) This is a common abuse: It's annoying to have to write $X + (X^2)$, $\overline{X}$, or give a name to the quotient map, so one instead refers to the image of $X$ in $\mathbf{C}[X]/(X^2)$ as $X$ again and hopes that no confusion results — unfortunately for the beginner, it inevitably does. Given this, is it clear why $(X)$ is the unique maximal ideal of this quotient?