In a local ring, the maximal ideal $\mathfrak{m}$ is principal $\implies \dim_k(\mathfrak{m}/\mathfrak{m}^2)\leq 1$

Question

This is Proposition 8.8, $ii)\implies iii)$ in Atiyah and Macdonald and it says there that this is clear.

It isn't for me though. How would I prove this?

Answer 1

The maximal ideal $m$ is principal implies that it is generated by one element whose class is a generator of $m/m^2$.

Answer 2

A system of generators of $\mathfrak m$ gives rise to a system of generators of the $k$-vector space $\mathfrak m/\mathfrak m^2$. Thus, if $\mathfrak m$ is principal, $\mathfrak m/\mathfrak m^2$ has a single generator (possibly $0$).