# Localization of self dual algebra over a ring

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?