Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are:

Is $\inf\{i\in \mathbb N \cup \{0\}\cup \{\infty\}\mid Ext^i_R(R/I,R)\neq 0\}=0?$

In the case of $S := k[x_1, \cdots, x_n]/(x_1^1, \cdots , x^n_n),$ and $J:=(x_1, \cdots, x_n)$, there is, (If I'm not mistaken), an affirmative answer because of Rees' theorem (B-H,1.2.5). But I've not any idea about the questions above.

What about: $\inf\{i\in \mathbb N \cup \{0\}\cup\{\infty\}\mid \lim_{\to_n} Ext^i_R(R/I^n,R)\neq 0\}=0?$

thank you.