Let $L$, $N$, and $M$ be right $R$-modules and let $\widehat{L}=\mathrm{Hom}_{\mathbb{Z}}\left(L,\mathbb{Q}/\mathbb{Z} \right)$ ($\widehat{N}$ and $\widehat{M}$ are defined analogously). Show that the following conditions are equivalent for the exact short sequence $$0 \rightarrow L \xrightarrow{f} M \xrightarrow{g} N \rightarrow 0$$
$(a)$ The sequence is pure.
$(b)$ For any finitely presented module $F$ and any morphism $F \xrightarrow{\alpha} N$, there exists a morphism $F \xrightarrow{\beta} M$ such that $g \circ \beta = \alpha$.
$(c)$ The sequence $0 \rightarrow \widehat{L} \rightarrow \widehat{M} \rightarrow \widehat{N} \rightarrow 0$ is pure.
$(d)$ The sequence $0 \rightarrow \widehat{L} \rightarrow \widehat{M} \rightarrow \widehat{N} \rightarrow 0$ splits.
The hint is to use the fact that the $Hom$ and tensor functors are adjoint functors, that is $Hom(A \otimes B, C) \cong Hom(A,Hom(B,C))$, to prove $(a) \Longleftrightarrow (c)$ and $(a) \Longleftrightarrow (d)$ and to use the fact (without proof) that $Hom_{S}(Hom_{R}(M,N),E) \cong M \otimes Hom_{S}(N,E)$ whenever ${}_{R}N_{S}$, $M_{R}$ is finitely presented and ${}_{S}E$ is injective to show that $(b) \Longleftrightarrow (c)$
I was able to show that $(a) \Longleftrightarrow (c)$ using the hint and $(d) \Longrightarrow (c)$ is clear since every splitting sequence is pure exact. However, I wasn't able to show that $(a) \Longrightarrow (d)$ and I don't see how to use the other hint to show the equivalence $(b) \Longleftrightarrow (c)$.
Any hints or ideas would be greatly appreciated. Thanks in advance.