ring definition(s)

[Creighton Dement]

Dear Seqfans, 

A sci.math thread recently mentioned the difference in the definition of
a ring given at Mathworld http://mathworld.wolfram.com/Ring.html and
Wikipedia http://en.wikipedia.org/wiki/Ring_theory  : the former
specifies that multiplication defined for a ring not necessarily be
associative; the latter specifies not only that multiplication be
associative, but that it must also have a unit (this part of the
definition seems unusual to me). 

If that is the case, shouldn't sequences such as
http://www.research.att.com/~njas/sequences/A037292
(Nonassociative rings with n elements) specify the definition being
used? 

For the record, I consulted with my "Handbook of Discrete and
Combinatorial Mathematics" which gives the definition of ring I am used
to:

ring: an algebraic structure (R, +, *) where R is a set closed under two
binary operations + and *, (R, +) is an abelian group, R satisfies the
associative law for multiplication, and R satisfies the left and right
distributive laws for multiplication over addition. 

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.