ring definition(s)

[Ed Pegg Jr]

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

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
>