Isomorphism between $R$-algebra $RG$ and $RG^{\ast}$

Question

Let $R$ be a commutative ring, $G$ be a finite abelian group. Consider a group ring $RG$ as an $R$-coalgebra. Is it true that $RG\simeq RG^{\ast}$ as an $R$-algebra? If the answer is true, please tell me the homomorphism that makes $RG\simeq RG^{\ast}$. Thank You

Answer 1

No, not in general. The dual algebra $RG^*$ is always canonically isomorphic to the ring of functions $R^G$ (with the algebra operations defined pointwise), but $RG$ may not be isomorphic to a product of copies of $R$. For instance, if $R=\mathbb{F}_2$ and $G$ is cyclic of order $3$, then $RG$ is clearly not isomorphic to $\mathbb{F}_2^3$ since $RG$ contains a unit of order $3$ and $\mathbb{F}_2^3$ does not.