Timeline for Does every finite abelian $p$-group $G$ admit a local ring structure with residue field of the same rank as $G$?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2017 at 20:25 | comment | added | Taras Banakh | It seems that I have found a good reference in the book "Finite Commutative Rings and their Applictions" of G.Bini and F.Flamini (researchgate.net/publication/…). Such rings are called Galois rings and are denoted by $G(p^k,r)$. So, thanks to all for the fruitful discussion. | |
| Mar 26, 2017 at 17:43 | comment | added | Taras Banakh | Thank you. But I am not an algebraist, so am not very fluent in all these theories. Maybe one of these two papers contains a required reference? projecteuclid.org/download/pdf_1/euclid.rmjm/1250131150 and another one is R.Gilmer, Zero-divisors in commutative rings, Amer. Math. Monthly, 93:5 (1986) 382--387? | |
| Mar 26, 2017 at 9:23 | comment | added | Jeremy Rickard | @TarasBanakh I'm not sure of a reference, but I think you can always find an unramified degree $n$ extension $\mathcal{O}$ of the $p$-adic integers $\mathbb{Z}_p$ and then take $G=\mathcal{O}/p^k\mathcal{O}$. | |
| Mar 25, 2017 at 19:05 | comment | added | Taras Banakh | Thank you very much for the answer (which was opposite to what I expected). Could you give me a reference to the fact that each group $(\mathbb Z/p^k\mathbb Z)^n$ is isomorphic to the additive group of a local ring whose residue field has cardinality $p^n$? | |
| Mar 25, 2017 at 13:35 | vote | accept | Taras Banakh | ||
| Mar 25, 2017 at 11:36 | history | answered | Jeremy Rickard | CC BY-SA 3.0 |