Timeline for Combinatorics of projective planes over commutative rings
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 10, 2021 at 16:43 | comment | added | Ben McKay | Have you considered the theory of ternary rings? | |
| Jun 3, 2021 at 6:23 | comment | added | MvG | @Max, that comment of yours sounds like an answer to me. It doesn't preclude getting a plane by some other bizarre construction. I'd say chances are low but disapproving the possibility would be hard. So I don't expect we get something significantly more complete than what you offer. | |
| Jun 1, 2021 at 13:38 | comment | added | Max Horn | What do you mean with a projective plane over an arbitrary commutative ring $A$? If you take the usual approach of considering rank 1 "subspaces" (here: submodules) of $A^3$ as points, and rank 2 "subspaces" as lines (I am being a bit sloppy for brevity), then this yields a projective plane in the axiomatic sense if and only if $A$ is a field: indeed, if $A$ isn't a field, then it admits a proper nontrivial ideal $I$, and you get "points" such as $\{ (x,0,0) : x\in I \}$ which are properly contained in other "points" (here: $\{ (x,0,0) : x\in A\}$, and they are not collinear via a unique line. | |
| Jun 1, 2021 at 11:38 | history | edited | THC | CC BY-SA 4.0 |
added 93 characters in body
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| Jun 1, 2021 at 11:19 | comment | added | THC | @abx: Oops, I messed up mu question. I will re-phrase it. | |
| Jun 1, 2021 at 3:22 | comment | added | abx | I probably don't understand the question, anyway: the set of $k$-rational points in $P$ is the classical projective plane over $k$, hence obviously an axiomatic projective plane. | |
| May 31, 2021 at 21:27 | history | asked | THC | CC BY-SA 4.0 |