Let $R$ be a commutative ring with identity and $A$ an $R$-algebra that is finitely generated, projective $R$-module and has identity. Furthermore, assume that $A$ and $Hom_R(A,R)$ are isomorphic as $A$-modules. For $\varphi\in Hom_R(A,R)$ and $a\in A$, $a\varphi(x)=\varphi(ax)$.

If for an element $a\in A$ one takes the localization $A_a$, we have that $A_a\cong Hom_R(A,R)\otimes_A A_a$. My question is, how can I think of the elements in $Hom_R(A,R)\otimes_A A_a$?

If $Hom_R(A,R)$ is generated by $\eta:A\rightarrow R$ as an $A$-module and $a\in \ker\eta$, can I define $a^{-1}\eta$ as a homomorphism?


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