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No. Let $A=\mathbb{Z}[1/2]/\mathbb{Z}$. Let $R\to S$ be $\mathbb{Z}\to \mathbb{Z}_2\times \mathbb{Z}_3$, where $\mathbb{Z}_n$ denotes $\mathbb{Z}[1/n]$.
Then $Ann_{\mathbb{Z}}A=0$. But $(1,0)\in S$ kills $A\otimes_{\mathbb{Z}}S$.
Let $A=\mathbb{Z}[1/2]/\mathbb{Z}$. Let $R\to S$ be $\mathbb{Z}\to \mathbb{Z}_2\times \mathbb{Z}_3$.
Then $Ann_{\mathbb{Z}}A=0$. But $(1,0)\in S$ kills $A\otimes_{\mathbb{Z}}S$.
No. Let $A=\mathbb{Z}[1/2]/\mathbb{Z}$. Let $R\to S$ be $\mathbb{Z}\to \mathbb{Z}_2\times \mathbb{Z}_3$, where $\mathbb{Z}_n$ denotes $\mathbb{Z}[1/n]$.
Then $Ann_{\mathbb{Z}}A=0$. But $(1,0)\in S$ kills $A\otimes_{\mathbb{Z}}S$.
