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May 1st, 2019, 10:17 PM   #1
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Definition of Formal power series in m indeterminates over R

I could not understrand the following definition for formal power series over $m$ indeterminates, over the commutative ring $R$:

*I do understand:*

We set $R[\![X_{1},...,X_{m}]\!]:=(R^{(\mathbb{N}^{m})},+,.)$,
where $+$ and $.$ are as in:
$(p+q)_{\alpha}:=p_{\alpha}+q_{\alpha}$, $\alpha\in\mathbb{N}$ and $(pq)_{\alpha}:=\sum_{\alpha<\beta}$ $p_{\beta}q_{\alpha - \beta}$, $\alpha\in\mathbb{N}$.

*But I get stuck in the following definition:*


Set $X:=(X_{1},...,X_{m})$ and , for $\alpha\in\mathbb{N}$, denote by $X^{\alpha}$ the formal power series (that is, the function $\mathbb{N}^m\rightarrow R$) such that:

$X_{\beta}^{\alpha}:=\lbrace 1$ for $\beta = \alpha,0$ for $\beta\ne\alpha\rbrace$, $\beta\in\mathbb{N}$

Then each $p\in R[\![X_{1},...,X_{m}]\!]$ can be written uniquely in the form $p=\sum_{\alpha\in\mathbb{N}^m}p_{\alpha}X^{\alpha }$

I could not build an example to visualize this, unlike the case for $m=1$ that was clear: by making $X=(1,0,...,0,..), X^2=(0,1,...,0,..)$ and so on.
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