Re: 2 rings with a special property
From: Robert Israel (israel_at_math.ubc.ca)
Date: 06/30/04
- Next message: mina_world: "Fields medal"
- Previous message: Robert Israel: "Re: fractal spirals"
- In reply to: Bessel: "2 rings with a special property"
- Next in thread: William Elliot: "Re: 2 rings with a special property"
- Messages sorted by: [ date ] [ thread ]
Date: 30 Jun 2004 05:45:29 GMT
In article <ce759023.0406291754.4e1bb810@posting.google.com>,
Bessel <crypto_170@hotmail.com> wrote:
>I want to find two rings R1, R2 and a homomorphism f: R1-->R2 between
>the two rings. I need some special properties:
>1. R1 should have many ideals
>2. Kernel of f should not look too "special" in any way.
>I.e. for example if we are dealing with matrices and the kernel of
>homomorphism is such that last column or last row is all zeroes, then
>it's not quite satisfactory because then it looks "special" as opposed
>to other regular elements which don't have this 0s property.
>3. I also would like |ker f|/|R1| to be fairly small.
I assume |.| is cardinality, so you're dealing with finite rings.
ker f of course will be an ideal. So I'd start with a large ring
with many small ideals, let J be a randomly-chosen small ideal, and
let f be the quotient map R1 -> R1/J. So ker f = J, and its elements
won't be any more "special" than the elements of any other small ideal.
For example, let R1 = (Z_2)^n with coordinatewise operations (so
(xy)_j = x_j y_j), S a nonempty subset of {1,...,n} (typically
with |S| about n/2), J = {x in R1: x_k = 0 for k in S}. You can
also identify f as the restriction map of R1 to
(Z_2)^(complement of S). Then |ker f|/|R1| = 2^(-|S|).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
- Next message: mina_world: "Fields medal"
- Previous message: Robert Israel: "Re: fractal spirals"
- In reply to: Bessel: "2 rings with a special property"
- Next in thread: William Elliot: "Re: 2 rings with a special property"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
|
