Divided power algebra
= Divided power algebra
{title2=$\Gamma_R(a)$}
For an even-degree generator, take a free graded $R$-module with basis $a_k$, $k\geq0$, and multiplication $a_i a_j=\binom{i+j}{i}a_{i+j}$, 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 $a_1$.