The divided polynomial algebra over a commutative ring $R$ in a symbol $X$ of degree $k$ is defined as the free $R$-algebra on symbols $X_1=X, X_2, X_3,\dotsc$ of degree $k$, $2k$, $3k,\dotsc$ modulo the relation $X_iX_j={i+j \choose j}X_{i+j}$. Now these symbols are naturally the divided powers $\gamma_i(X)$ of $X$ and the relation $\gamma_i(X)={i+j \choose j}\gamma_{i+j}(X)$ holds.

My problem: This equation, however, needs to hold for all elements of the divided polynomial algebra, e. g. for $X_2$ and I can't see why it does. Second, what about other requirements of divided powers such as $\gamma_i(X_iX_j)=\gamma_i(X_i)\gamma_j(X_j)$? How is $\gamma_i$ even defined for elements $\neq X$?

All difficulties resolve for $R\subset \mathbb{Q}$ so an answer for the general situation would be very much appreciated. Thank you!


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