Let $R$ be a commutative ring, and $p$ be a prime ideal. Let $a_1, ... a_n \in p$. I'm looking for a criterion that the images of $a_1, ... a_n$ in $pR_p/(pR_p)^2$ are spanning or linearly independent, using only $R$ and $p$ itself (without using any localization or quotient).
Here are my guesses:
- $a_1, ... a_n$ span iff $(a_1, ... a_n)$ generate $p$ as an ideal.
- $a_1, ... a_n$ are linearly independent if, for all $b_1, ... b_n$ with $\sum_{i=1}^n a_i b_i \in p^2$, $b_i \in p$ for all $i$.
I'm trying to prove these criterion. If $a_1, ... a_n$ generate then they are a spanning set. I think that the converse uses Nakayama's lemma somehow.
For the linear independence one, suppose $c_i \in R_p / pR_p$ is such that $\sum c_i a_i \in (pR_p)^2$. Clearing denominators, we can assume $c_i \in R$. But I'm not sure if $a/1 \in (pR_p)^2$ implies $a \in p^2$.