Let $A$ be an order, i.e. a commutative ring of which the additive group is isomorphic to $\mathbb{Z}^n$ for a certain non-negative integer $n$. Show that there exists an embedding $A^{\times}_{\text{tor}}\ \hookrightarrow\ (A_{\text{red}})^{\times}_{\text{tor}},$ where $A^{\times}_{\text{tor}}$ is the group of torsion units of $A$, and $A_{\text{red}}=A/\sqrt{0_A}$ is the reduced ring of $A$.
Edit: In response to a reply which seems to have been removed; I understand that the quotient map $A\rightarrow A_{\text{red}}$ restricts to a group homomorphism $A^{\times}_{\text{tor}}\rightarrow(A_{\text{red}})^{\times}_{\text{tor}}$, but I am unable to show that this map is injective.