Quotients of local rings

Question

I have the following commutative algebra question. It came up at the time of reading local rings of schemes and their tangent spaces.

Let $R$ be a local ring with unique maximal ideal $m$. Then is it true that $R/m^2 \simeq R/m \oplus m/m^2$?

Thank you.

Answer 1

No. Let $\Bbb{Z}_{(p)}$ be the localization of $\Bbb{Z}$ at the prime ideal $(p)=p\Bbb{Z}$. $$ \begin{split} \Bbb{Z}_{(p)}/p^2\Bbb{Z}_{(p)}\simeq \Bbb{Z}/p^2\Bbb{Z}\\ \Bbb{Z}_{(p)}/p\Bbb{Z}_{(p)}\simeq \Bbb{Z}/p\Bbb{Z}\\ (\Bbb{Z}_{(p)}/p\Bbb{Z}_{(p)})/(\Bbb{Z}_{(p)}/p^2\Bbb{Z}_{(p)})\simeq \Bbb{Z}/p\Bbb{Z}\\ \Bbb{Z}/p^2\Bbb{Z}\not\simeq\Bbb{Z}/p\Bbb{Z}\oplus\Bbb{Z}/p\Bbb{Z} \end{split} $$