Timeline for Do there exist general conditions for cyclicity of unit groups of quotient rings (generalizations of the primitive root theorem)?
Current License: CC BY-SA 4.0
8 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| May 14, 2020 at 10:25 | history | edited | darij grinberg | CC BY-SA 4.0 |
added 2 characters in body
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| May 12, 2020 at 15:22 | vote | accept | Daniel Santiago | ||
| Apr 13, 2020 at 23:03 | comment | added | R. van Dobben de Bruyn | A commutative ring $R$ with $R^\times = \mathbf Z$ necessarily has characteristic $2$, since $-1$ has order $2$ unless $-1 = 1$. It is necessarily reduced since $1+x$ has order $2$ if $x \neq 0$ but $x^2 = 0$. An example is $R = \mathbf F_2[x,x^{-1}]$, which is the group ring $\mathbf F_2[\mathbf Z]$. Any such $R$ contains $\mathbf F_2[x,x^{-1}]$. Whenever $R$ is an example, so is the polynomial ring $R[x_i\ |\ i \in I]$ for any set $I$. Classifying such rings is almost certainly impossible (as is for example writing down all $\mathbf C$-algebras with $R^\times = \mathbf C^\times$). | |
| Apr 13, 2020 at 12:01 | comment | added | Daniel Santiago | This is amazing! Thank you so much! Out of curiosity for the infinite case, does there exist a ring $R$ with non-zero integral ideal $I$ such that $(R/I)^{\times} \cong \mathbb{Z}$? I think that such an example could come from group rings but I'm not certain | |
| Apr 13, 2020 at 5:17 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Small corrections and clarifications.
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| Apr 13, 2020 at 5:01 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Small clarification of used notation.
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| Apr 13, 2020 at 4:56 | history | answered | R. van Dobben de Bruyn | CC BY-SA 4.0 |