Corollary 2

Let M be a finitely generated A-module. For S^{-1}M = 0, it is necessary and sufficient that there exists s \in S such that sM = 0.

Without any conditions on M, clearly the relation sM = 0 for some s \in S implies S^{-1}M = 0. Conversely, suppose that S^{-1}M = 0 and let (m_i)_{1 \leq i \leq n} be a system of generators of M; the m_i/1 generate the S^{-1}A-module S^{-1}M, hence to say that S^{-1}M = 0 amounts to saying that m_i/1 = 0 for 1 \leq i \leq n; by proposition 4 there exist $s_i \in S$ such that $s_i m_i = 0$ and, taking $s = s_1 s_2 \cdots s_n$, $s m_i = 0$ for all $i$ and hence $sM = 0$.