Re: ISO example of a concrete "non-arithmetic" ring
- From: "John Coleman" <jcoleman@xxxxxxxxxxxxxx>
- Date: 7 Feb 2006 18:39:10 -0800
kj wrote:
I know of examples of groups that can be readily understood by a
lay person without requiring any use of numbers and arithmetic
(e.g. the group of permutations of a finite set). But I can't
think of any "non-arithmetic" examples of rings. Does anyone know
of one?
(This is strictly for my own edification. It seems to me that the
notion of a ring is a fundamentally arithmetic construct; i.e.
without arithmetic rules of combination, one has no rings. Is this
impression correct?)
Your notion of "arithmetical" isn't well-defined, but there is a sense
in which your impression isn't 100% wrong. Rings are abelian groups
with respect to addition, and abelian groups are often (always in the
finitely-generated case) built up from things isomorphic to the
integers or the integers mod n for various n. I don't recall how
"nonarithmetical" infinite abelian groups can get, but the sorts of
intuitions you get from looking at Z and its homomorphic images only
goes so far.
.
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