(Solved): 5.2. Let :AB be a homomorphism of R-modules. Prove that is an epimorphism if the induced ...

5.2. $\neg$ Let $\alpha :A\to B$ be a homomorphism of $R$-modules. Prove that $\alpha$ is an epimorphism if the induced ${\mathrm{map}}^{24}{\alpha }^{\ast }:{\mathrm{Hom}}_R(B,N)\to {\mathrm{Hom}}_R(A,N)$ is injective for all $R$-modules $N$ and $\alpha$ is a monomorphism if ${\alpha }^{\ast }$ is surjective for all $N$. Prove that the converse to the first statement holds and the converse to the second statement does not hold. However, show that if $\alpha$ admits a left-inverse, then ${\alpha }^{\ast }$ is surjective for all $N.[5.3,6.3]$