ring definition(s)

[p.j.cameron at qmul.ac.uk]

On the question of the identity element in a ring: There is a
substantial minority view that a ring should have an identity.
This has some positive advantages: if a ring has an identity, then the
commutative law for addition can be deduced from the other axioms.
People who adopt this convention have even coined the term "rng" for a
ring which does not necessarily have an identity!

In my view the disadvantages outweigh the advantages. Among them:
1. An ideal isn't a subring (unless it is the whole ring)
2. Some important rings from functional analysis, such as the ring of
(lebesuge-)integrable functions on R, or the ring of functions with
compact support, are excluded.

Peter Cameron.

On Sat, Jan 28, 2006 at 11:12:44PM -0500, franktaw at netscape.net wrote:
> Certainly the most common definition of a ring does require 
> associativity, and does not require a unit. However, this is not 
> universal. For example, my copy of Birkhoff and MacLane's "Modern 
> Algebra" (The MacMillen Company, 1969) does require a unit.
> 
> I have also seen a definition of subring which requires that, if the 
> ring has a unit, the subring has the same unit. This raises the bizarre 
> possibility that for three rings R, S, and T, R is a subring of S, 
> which is a subring of T, but R is not a subring of T.
> 
> Best to stick with the most standard definition, but do be aware that 
> there are others in use.  (A phrase like "non-associative ring" is 
> sufficient to indicate that the standard definition of ring is not 
> being used.)
> 
> -----Original Message-----
> From: Creighton Dement <crowdog at crowdog.de>
> 
>  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).
> 
> 
> 
> 
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