Multivariate formal power series

Remarks

The inner sum in the product is finite, since each $\alpha$ has finitely many decompositions $\alpha = \beta + \gamma$ in $\N^n$. For $n = 1$ this is the ring of formal power series in one variable, and $R[\![X_1, \dots, X_n]\!] \cong R[\![X_1, \dots, X_{n-1}]\!][\![X_n]\!]$. Example: in $\Z[\![X, Y]\!]$ the series $f = \sum_{k \ge 0} (X + Y)^k$ satisfies $(1 - X - Y) f = 1$, because the coefficient of each monomial of degree $k \ge 1$ cancels between $(X + Y)^k$ and $(X + Y)(X + Y)^{k-1}$. Non-example: $\sum_{k \ge 0} X^{-k} Y^k$ is not a formal power series in $X, Y$, since the exponents must lie in $\N^n$. A series is a unit if and only if its constant coefficient $a_{(0, \dots, 0)}$ is a unit of $R$.