AW: Re: ring definition(s)

[Creighton Dement]

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> Date: Mon, 30 Jan 2006 18:48:36 +0100
> Subject: Re: ring definition(s)
> From: Ed Pegg Jr <edp at wolfram.com>
> To: njas at research.att.com

> Actually, Neil, it's OEIS that gave me faith in non-associative rings,
> along with work by Conway on Octonions.
> 
> http://www.research.att.com/~njas/sequences/A037292
> 
> Although I believe OEIS and Conway are correct, non-associative rings
> don't seem to be popular.  My check of 10 various references all
> declare rings to always be associative, and that's what most college
> texts say.  So, Eric and I will change the definition to require
> associativity by definition,
> but still mentioning the existence of non-associative rings.
> 
> Requiring a multiplicative identity is only in 2 books I checked.  All
> others
> indicate that is an optional quality.
> 
> Ed Pegg Jr

I'm no expert on those beautiful octonions, but it would appear that at
least part of what makes them interesting is that they are "among the
least non-associative, of all non-associative rings" (or quasi-ring)
i.e. we're "merely" talking about a possible  a*(b*c) = -(a*b)*c and not
some totally spaced-out result.  (more below)


> N. J. A. Sloane wrote:
> 
> > Responding to Creighton's question:  one always requires
> > associativity for multiplication and addition in a ring.
> > 
> > But the existence of a multiplicative unit is not required.
> > 
> > The books by Lam are the canonical references,
> > but there are a huge number of others.
> > Don't believe anything you read on the web - except
> > on my home pages and the OEIS!
> > 
> > Neil
> > 

Thanks to everyone. To me, it seems important  - at least for A037292 -
to decide whether the reference is to  "Number of non-communative rings
with a unit" or "Number of non-communitive rings with or without a
unit". Following the cross-reference sequence to
http://www.research.att.com/~njas/sequences/A027623
gives a definition which one might assume is being applied here as well:

Here a ring means (R,+,*): (R,+) is abelian group, * is associative,
a*(b+c) = a*b+a*c, (a+b)*c = a*c+b*c. Need not contain "1", * need not
be commutative.

I wonder (apologies if this is a stupid question)... is "Number of
non-communative rings without a unit" well-defined?

Sincerely, 
Creighton  

-It's a shame when the girl of your dreams would still rather be with
someone else when you're actually in a dream.