Timeline for Categorical definition of the ideal product within the category of rings
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Nov 23, 2021 at 21:26 | vote | accept | Martin Brandenburg | ||
| Nov 23, 2021 at 21:26 | comment | added | Martin Brandenburg | What is implicit here: To arrive at a categorical characterization, we also need to remark that $C_K$ is the kernel pair of $R \to R/K$ in $\mathbf{CRing}$, and that $R \to R/K$ is the coequalizer of its kernel pair --- so this explains why we can work with congruences instead of quotients here. | |
| Nov 23, 2021 at 21:23 | comment | added | Martin Brandenburg | I only see this answer now (I don't check notifications anymore here). Awesome, finally this problem gets solved :). Thanks a lot anonymous user. | |
| Jan 3, 2017 at 18:12 | comment | added | user13113 | @R.vanDobbendeBruyn: Right; the overall idea was inspired by Martin's suggested approach by unitalization. | |
| Jan 3, 2017 at 18:12 | history | edited | user13113 | CC BY-SA 3.0 |
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| Jan 3, 2017 at 16:54 | comment | added | R. van Dobben de Bruyn | And I suppose that the point of this answer is that $C_K$ and all other objects involved are in fact rings (rather than mere modules). | |
| Jan 3, 2017 at 15:45 | history | edited | user13113 | CC BY-SA 3.0 |
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| Jan 3, 2017 at 15:41 | comment | added | user13113 | @R.vanDobbendeBruyn: Hrm. In my scratchwork I had used $R \oplus K$ throughout, and only changed to $C_K$ in the final revision, so the choices were indicated by the notation. In the end they only matter up to isomorphism, so I don't think it necessary to bog it down with the details of a particular choice, but I suppose it's short enough that it shouldn't hurt much to fill in more details. | |
| Jan 3, 2017 at 15:06 | comment | added | R. van Dobben de Bruyn | Your notation is very sloppy. Which of the two projections $p_i \colon C_J \to R$ do you use for $\pi_0$? How do you view $R$ as a subring of $C_J$ in the definition of $T$? Which 'evident isomorphism' $C_K \cong R \oplus K$ do you choose? You could take $(p_1,p_1-p_2)$, or $(p_2,p_1-p_2)$, or a sign variation thereof. | |
| Jan 3, 2017 at 6:45 | history | undeleted | user13113 | ||
| Jan 3, 2017 at 6:45 | history | edited | user13113 | CC BY-SA 3.0 |
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| Jan 3, 2017 at 6:25 | history | deleted | user13113 | via Vote | |
| Jan 3, 2017 at 6:02 | history | edited | user13113 | CC BY-SA 3.0 |
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| Jan 3, 2017 at 5:56 | history | answered | user13113 | CC BY-SA 3.0 |