Divided power algebra (${\Gamma }_R(a)$)

For an even-degree generator, take a free graded $R$-module with basis $a_k$, $k\ge 0$, and multiplication $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$, with $a_0=1$ and $|a_k|=k|a|$. Associativity follows from the multinomial identity. Over $\mathbb{Z}$, $a_1^k=k!a_k$, so the algebra need not be polynomial on $a_1$.