The theorem 4.10 iii) in the page 204 of Hungerford says:
if $R$ is a division ring and $0\rightarrow A\stackrel{\theta}{\rightarrow}B\stackrel{\zeta}{\rightarrow}C\rightarrow 0$ is a short exact sequence of left vector spaces, then $0\rightarrow C^*\stackrel{\overline{\zeta}}{\rightarrow}B^*\stackrel{\overline{\theta}}{\rightarrow}A^*\rightarrow 0$ is a short exact sequence of right vector spaces where $A^*,B^*$ and $C^*$ denote the dual space. The proof is an exercise, but the suggestion is not clear. I thought of using the theorem 4.6 iii) of page 202
If $0\rightarrow A\stackrel{\theta}{\rightarrow}B\stackrel{\zeta}{\rightarrow}C\rightarrow 0$ is a short exact sequence of R-modules, then $0\rightarrow \text{Hom}_R(C,J)\stackrel{\overline{\zeta}}{\rightarrow}\text{Hom}_R(B,J)\stackrel{\overline{\theta}}{\rightarrow}\text{Hom}_R(A,J)\rightarrow 0$ is an exact sequence of abelian groups where J is a injective module.
But then, I need prove that if $R$ is a division ring then $R$ is injective. This is true? How do I prove this or what is the counterexample? What is the strategy by proof the theorem 4.10 iii?