It is clear that proving that the power series $\sum a_nx^n$ is a ring implies that the ordinary generating functions are also a ring.
My doubt is, the exponential generating $\sum a_n\frac{x^n}{n!}$ functions are also a ring? How do I justify this knowing the above?
Perhaps what you are really asking is the following.
Suppose given a ring $R$. The set of all sequences $(a_0,a_1,\dots)$ of elements in $R$ forms a vector space over $R$. Let multiplication be defined by $a\star b = c$ where $c_n := \sum_{k=0}^n a_kb_{n-k}.$ The vector space now becomes a ring isomorphic to the ring of formal power series in $R$ given by the ordinary generating function map.
Now define another multiplication on the vector space by $a\ast b = c$ where now instead $c_n := \sum_{k=0}^n {n\choose k}a_kb_{n-k}.$ The vector space now becomes a ring isomorphic to the ring of formal power series in $R$ except the mapping is by exponential generating function.