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What does $\langle2\rangle$ mean?

I encountered it in the context of $\mathbb{Q}_2$ and it seemed most likely to mean something like $\{2^z:z\in\mathbb{Z}\}$ from what I could deduce but I've been unable to find a reference to it.

it's a hire car baby
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  • It is the ideal of the $2$-adic rationals generated by $2$. – Mathematician 42 Aug 29 '17 at 08:26
  • @Mathematician42 thanks; and does that correspond with my interpretation? I see ideals include the multiples of e.g. some integer which would imply ${6,10,12,\ldots}$ are also elements - the even numbers. – it's a hire car baby Aug 29 '17 at 08:29
  • p.s. does the set I wrote have a name? – it's a hire car baby Aug 29 '17 at 08:33
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    Context will tell. $\langle 2\rangle$ is some substructure generated by $2$. It could be either A) the multiplicative subgroup generated by $2$, B) the additive subgroup generated by $2$, C) an ideal (of a ring "obvious" from the context) generated by $2$. In the case of $\Bbb{Q}_2$ my instincts suggest that interpretation A is the most likely (the one you inferred also). This is because for B I would write $2\Bbb{Z}$ and for C I would write $2\Bbb{Z}_2$ (assuming that the ring of 2-adic integers is the ring whose ideals we would be interested in). – Jyrki Lahtonen Aug 29 '17 at 08:39
  • @Mathematician42 it would seem $\langle2\rangle$ contains ${x:\lvert x\rvert_2\leq 1}$ - is that correct? – it's a hire car baby Aug 29 '17 at 08:40
  • I guess it might be possible to write $2^{\Bbb{Z}}$ for the multiplicative generated by $2$, but that could easily be confused with the powerset notation. – Jyrki Lahtonen Aug 29 '17 at 08:43
  • @JyrkiLahtonen thanks for the general answer; I now know how to interpret it both in general and this case. So in the context it would be $A={\ldots,2,4,8,\ldots}$ (and incidentally it was you who wrote it) – it's a hire car baby Aug 29 '17 at 08:48
  • In that case you should obviously give less weight to my opinion here. I didn't remember the occasion I used it, but anyway, the opinion I wrote here is clearly colored by my personal preferences. As is my past use. – Jyrki Lahtonen Aug 29 '17 at 08:51
  • $<2>$ is the intersection of all subgroups of $\mathbb Q_2$ containing $2$. It's usually referred to as the subgroup generated by 2. – Steven Alexis Gregory Aug 29 '17 at 09:05
  • The subgroups of $\mathbf Q_2^\times$. – Bernard Aug 29 '17 at 09:56
  • @Mathematician42: It is not an ideal. There aren't so many ideals in a field. – Bernard Aug 29 '17 at 09:57
  • @Bernard: You're right, I didn't really think that through. – Mathematician 42 Aug 29 '17 at 15:10

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