What does $\mathbb{Z}[[t]]$ mean? Why are there double square brackets?

I can't search through Google, because I can't search Latex.

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    $\begingroup$ I think it's the ring of formal power series with coefficents from $\mathbb{Z}$, see en.wikipedia.org/wiki/Formal_power_series $\endgroup$ – Mikko Korhonen Sep 2 '12 at 16:19
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    $\begingroup$ You actually can search LaTeX: latexsearch.com $\endgroup$ – huon Sep 2 '12 at 16:22
  • $\begingroup$ If this is from a book, it might have an list of symbols in the back that you can check. $\endgroup$ – Jair Taylor Sep 2 '12 at 17:10

That is the ring of formal power series in $t$ with integer coefficients, i.e., of $$\sum_{n=0}^\infty a_nt^n,$$ with $a_n\in\Bbb Z$, componentwise addition, and multiplication appropriately defined.

The double brackets distinguish it from $\Bbb Z[t]$, which is the ring of polynomials in $t$ with integer coefficients. We can always evaluate the members of $\Bbb Z[t]$ for any complex value of $t$, but we generally can't evaluate members of $\Bbb Z[[t]]$ for $t\neq 0$. To my mind, the double bracket is a reminder that we need to leave the $t$ alone, and not worry about evaluation.

  • $\begingroup$ (IMO there is a sense in which can view "evaluation at $t=a$" to be the quotient map $\Bbb Z[[t]]\to \Bbb Z[[t]]/(t-a)$, though this is cheating and doesn't send everything all the way down to $\Bbb Z$.) $\endgroup$ – anon Sep 2 '12 at 17:08
  • $\begingroup$ Interesting point! $\endgroup$ – Cameron Buie Sep 2 '12 at 17:11
  • $\begingroup$ I laughed. I'm going to steal this saying. Many thanks. "just leave $t$ alone" ha. $\endgroup$ – James S. Cook Sep 2 '12 at 18:21
  • $\begingroup$ Have at it, James! $\endgroup$ – Cameron Buie Sep 3 '12 at 1:47
  • $\begingroup$ @CameronBuie Is it ok to use the symbol $\mathbb{Z}\llbracket t\rrbracket$ in latex instead of $\mathbb{Z}[[t]]$ or is that wrong? $\endgroup$ – Pratyush Sarkar Mar 4 '13 at 14:51

If $A$ is any ring, the notation $A[[T]]$ stands for the ring of formal power series with coefficients in $A$, i.e. the ring whose elements are the expressions $$ a_0+a_1T+a_2T^2+a_3T^3+\cdots $$ with the obvious sum and product.


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