Q:

What is precise definition of $\Gamma_{\mathfrak{a}}(M)$?

For a module $M$ over a ring, $\mathfrak{a} \subseteq A$ any ideal we define $\Gamma_{\mathfrak{a}}(M) = \{m \in M: \mathfrak{a}^nm = 0$ for some $n > 0\}$. What exactly is meant by $\mathfrak{a}^n m = 0$ for some $n > 0$? Does it mean EVERY element in $\mathfrak{a}^n$ annihilates $m$ or THERE IS SOME element in $\mathfrak{a}^n$ that annihilates $m$?

A:

It means every element.$      $

