Is R/m a flat R-module?

Question

Let $(R,\frak m)$ be a commutative Noetherian local ring. Is $R/\frak m$ a flat $R$-module? Thanks.

Answer 1

From this answer you can learn that if $R/\mathfrak m$ is $R$-flat then $\mathfrak m=\mathfrak m^2$. Since $\mathfrak m$ is finitely generated it must be generated by an idempotent, that is, $\mathfrak m=(0)$ so $R$ is a field.