Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring

Closure follows from [L2]; the additive group laws, including the zero sequence and coefficientwise negatives, follow coefficient by coefficient from the additive group laws in $R$.

Distributivity follows by splitting each finite convolution sum, commutativity follows by reindexing $(i,j)$ as $(j,i)$ and using commutativity in $R$, and associativity follows by [L3] from the equality of the two finite sums $\sum_{i+j+k=n}a_i b_j c_k$; the sequence $1$ is a multiplicative identity because only the index-$0$ coefficient contributes. Finally $c(r+s)=c(r)+c(s)$, $c(rs)=c(r)c(s)$, and $c(1_R)=1$, so [L4] makes $c$ a unital ring homomorphism, while equality of constant sequences forces equality of their index-$0$ coefficients and makes $c$ injective.