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Is it possible to have ring isomorphisms between some subsets of size $s^k$ of Galois ring $\Bbb Z_2^{s^k}$ and the full Galois ring $\Bbb Z_s^k$?

azimut
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Turbo
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1 Answers1

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No, at least for $s\ne 2$, since this subset must be a subring of order/size s^k. In particular, it is an additive subgroup in the group $\Bbb Z_2^{s^k}(+)$. Hence $s^k$ must divide $2^{s^k}$.

Boris Novikov
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  • Do you want to define other operations on $\Bbb Z_2^{s^k}$? Anyway my resoning is true, since you will have other ring of the same order. – Boris Novikov Sep 07 '13 at 15:00
  • I cannot take this as an answer. It is not rigorous enough. " since this subset must be a subring of order/size .." may be there is a clever way to force this for just the chosen subset leaving this subset closed under special algebraic operations.

    However, I understand what you are saying about divisibility of order.

     – Turbo Sep 07 '13 at 15:00
  • Yes may be there is a non-natural operation in the subset that is just compatible with $\Bbb Z_s^k$ while the operation while extendible to the whole ring produces disjoint closed sets under the algebraic operation. May be the $0$ in $\Bbb Z_s^k$ need not be mapped to traditional $0$ in the other ring? That is, the new $0$ and $1$ in the subset may not act as a $0$ and $1$ for the whole ring. – Turbo Sep 07 '13 at 15:02
  • There could be other automorphisms of $\Bbb Z_s^k$ other than the standard one as well. – Turbo Sep 07 '13 at 15:08
  • I think you have to formulate the question more correctly. – Boris Novikov Sep 07 '13 at 15:12