Let $J$ be an ideal of $R$ and $M$ an $R$-module. Suppose that $\mathrm{Ext}^{1}(R/J,M)=0$. Is it true that $\mathrm{Ext}^{n}(R/J,M)=0$ for $n\ge1$?

Question

Let $R$ be a commutative ring and $M$ be an $R$-module. We know that $Ext^{1}(R/I,M)=0$, for any ideal $I$ of $R$ if and only if $Ext^{n}(R/I,M)=0$, for any ideal $I$ of $R$ and any $n \geq 1$. Is this true for just one ideal?

Let $J$ be an ideal of $R$ and $M$ an $R$-module. Suppose that $\mathrm{Ext}^{1}(R/J,M)=0$. Is it true that $\mathrm{Ext}^{n}(R/J,M)=0$ for $n\ge1$? How to prove this?

Answer 1

Let $(R,\frak m)$ be a local ring and suppose that ${\rm depth}\, R=2$. Then obviously ${\rm Ext}_R^1(R/{\frak m},R)=0$ but ${\rm Ext}^2_R(R/{\frak m},R) \neq 0$.