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 Civic Duty
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Mar
24
comment Do principal divisors always have degree 0? (from Garrity)
@QiaochuYuan Oh, oops. That fixes it. Can you add it as an answer below?
Mar
24
asked Do principal divisors always have degree 0? (from Garrity)
Mar
19
comment How many combinations of permutation matrices are there?
There may be a recurrence relation you can find.
Mar
10
accepted A question about classifying conics, from Garrity's book
Feb
25
awarded  Civic Duty
Feb
5
revised Line bundles of the circle
"that that"
Feb
5
suggested approved edit on Line bundles of the circle
Jan
30
accepted Random Wolfram|Alpha identity related to $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$
Jan
30
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
@user26857 Good to know, thanks.
Jan
30
revised Random Wolfram|Alpha identity related to $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$
edited title
Jan
29
awarded  Notable Question
Jan
27
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
@user268578, is this why the localization you give is the "correct one"? Your method uses a set which has $(1,0)$ and $(2^k,1)$ as extra elements (which are not in the "real" multiplicative set). But the latter are all already invertible, so adding a new inverse makes no difference, and an inverse of $(1,0)$ is already going to be present, since if $r$ is the inverse of $(2,0)$, then $2r \times (1,0) = r \times (2,0) = 1$.
Jan
16
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
For anyone stumbling upon these comments: $(0,0)$ is not the multiplicative identity! :P
Jan
16
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
@VJP Why don't you add an answer? The hint was very helpful.
Jan
16
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
Um, I think then every element is zero. Is this justified? $$a = a\times 1 = a\times(r\times (2,0)) = a\times(r\times (0,0)/r) = 0$$Wikipedia seems to do something similar.
Jan
16
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
Quotients of the form $$\frac{(a,b)}{(2, 0)^n}?$$ That probably does not exclude $(1,0)$, since that can be remedied by adding a factor of $2$ to the numerator.
Jan
16
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
i.e. as a quotient of some "well-known" ring, etc.
Jan
16
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
@VJP Yes, I have. I haven't used it much besides proving that localizations at prime ideals are local rings. It seems that this ring is the localization of $\Bbb{R\times R}$ at the multiplicative set $$\{(1,1), (1,0), (2,0), (4,0), \ldots\}$$, but how do I describe that further?
Jan
16
comment Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$
Elementwise addition and multiplication.
Jan
16
asked Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$