Prove $\lambda:\text{Hom}(A,B)\otimes C\to\text{Hom}(A,B\otimes C)$ determined by $\lambda(f\otimes c)(a)=f(a)\otimes c$ is isomorphism, with following conditions are satisfied.
- Based ring $R$ is principal ideal domain.
- $A$ is free $R$-module.
- $C$ is finitely generated.
Of course, $\lambda$ is $R$-homomorphism. If we let $\eta:\text{Hom}(A,B)\times C\to\text{Hom}(A,B\otimes C)$ as $\eta(f,c)(a)=f(a)\otimes c$, then $\eta(f,c+rd)(\bullet)=f(\bullet)\otimes (c+rd)=f(\bullet)\otimes c+r(f(\bullet)\otimes d)=(\eta(f,c)+r\eta(f,d))(\bullet)$ holds. Also $\eta(f+rg,c)(\bullet)=(f+rg)(\bullet)\otimes c=(f(\bullet)+rg(\bullet))\otimes c=f(\bullet)\otimes c+r(g(\bullet)\otimes c)=(\eta(f,c)+r\eta(g,c))(\bullet)$. Therefore $\eta$ is $R$-bilinear, so induced map $\lambda:f\otimes c\mapsto\eta(f,c)$ is well-defined.
And also two $R$-modules $\text{Hom}(A,B)\otimes C, \text{Hom}(A,B\otimes C)$ are isomorphic when $A$ is free. Because if $A\cong\bigoplus_{i\in I}R$, then $\text{Hom}(A,B)\otimes C\cong\text{Hom}(\bigoplus_{i\in I}R,B)\otimes C\cong(\bigoplus_{i\in I}\text{Hom}(R,B))\otimes C\cong\bigoplus_{i\in I}B\otimes C\cong\bigoplus_{i\in I}(B\otimes C)$, and $\text{Hom}(A,B\otimes C)\cong \text{Hom}(\bigoplus_{i\in I}R,B\otimes C)\cong \bigoplus_{i\in I}\text{Hom}(R,B\otimes C)\cong\bigoplus_{i\in I}(B\otimes C)$.
But I cannot verify this homomorphism is isomorphism with above contitions. I know finitely generateed $R$-module is isomorphic with $\bigoplus_1^n R/(p_i)$ for some prime elements $p_i$, when $R$ is principal ideal domain. But I cannot go further.
Any help is welcomed. How can I prove $\lambda$ is isomorphism?
※ This problem is Exercies 1.8D of S-T Hu's book Introduction to Homological Algebra.