From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Fields and field-like structures
Date: 23 Jan 1998 21:35:23 GMT

In article <34C860CD.18E9@duro.upjs.sk>,
Gabriel SEMANISIN  <semanisin@duro.upjs.sk> wrote:

>	I am teacher of higher algebra at the university. I prepare some tasks
>for students, concerning an check whether given set with defined
>operations of "addition" and "subtraction" is field, and if not, whose
>of the axioms of field are valid in this structure. I am looking for
>some nice and computationally not very difficult examples of fields or
>similar structures; especially, I am interested in "almost fields", that
>means, the structures with two binary operations, in which every axiom
>of field is valid EXCEPT the one. Where on the Internet can I find such
>examples?

Well, there is a "Catalogue of Algebraic Systems" at
	http://www.math.usf.edu/algctlg/INDEX.html

With an eye towards the questions of real division algebras I wrote up
a FAQ which begins by discussing rings in general; you might begin at
	index/products.html
which assembles a couple of other related links.

Given the connection between coordinate rings and coordinate geometries, 
it shouldn't be surprising that a book such as
	Daniel Hughes, Fred Piper, "Projective Planes", Springer GTM 6
considers quite a few variations (a table on p154 compares
skew-, near-, quasi-, etc.- fields).

Of course it should be stressed to the students that the point of most
of these examples is to show that the axioms are independent; it can
hardly be claimed that these examples are all useful.

dave