# Both $R$ and $R/I$ are regular local rings

Let $R$ be a Noetherian local ring and $I$ is an ideal of $R$ such that both $R$ and $R/I$ are regular local rings. Could we deduce that $I$ is generated by an $R$-sequence?

I know that a noetherian local ring is regular if and only if its maximal ideal is generated by an $R$-sequence. Thanks for any cooperation.

## 1 Answer

From Bruns and Herzog, Proposition 2.2.4, we learn that if $R$ is a regular local ring, then $R/I$ is local regular if and only if $I$ is generated by a subset of a regular system of parameters. Now the conclusion follows.