Re: seeking definitions for commutative algebra

From: Stuart M Newberger (smnewberger_at_comcast.net)
Date: 01/18/05 

Date: 18 Jan 2005 04:08:48 -0800

Jyrki Lahtonen wrote:
> Stuart M Newberger wrote:
> > "Zariski O.and Samuel P.;Commuatative Algebra ,vol 1,"does not
assume
> > an identity in their definition of commutative ring .Starting on
page
> > 10 ,ring means commutative ring but in hypothesis where an identity
is
> > needed it is always explicitly referred to as a ring with
> > identity.Regards,Stuart M Newberger
> >
> Ok. Zariski & Samuel is a classic, I give you that. I
> learned whatever commutative algebra I know from Matsumura's
> "Commutative ring theory", where even the need to emphasize
> the fact that all the rings are assumed to have a 1 is moved
> out of the main body of text.
>
> May be it is just me, but I always thought the phrase "all
> the rings are assumed to have a 1" was often included for
> historical reasons only. Then again I learned my basic
> algebra from Jacobson, and his rings always have a 1. He
> does add a small subsection about "rngs" (=rings without
> unity). As a loyal disciple I absorbed the attitude that
> the burden of adding extra attributes to the concept of
> "ring" should be moved to the minority usage. So IMVHO
> there exist "rings" and also "rings without unity".
>
> What's the general feeling? Am I alone in thinking that
> the phrase "ring with unity" is only used because some
> old texts did it, but the shift is towards dropping the
> IMHO unnecessary clause "with unity"?
>
> Cheers,
>
> Jyrki Lahtonen

Ah yes ,Jacobson,that brings back memories.One of the first books I
came to own in the 60's was his Lectures in Abstract Algebra vol 1
Basic Concepts vol 2 Linear Algebra (and later appeared vol 3,Field
theory).Later
on he wrote a 2 volume work ,the first of which is called Basic Algebra
which is perhaps the one you are referring to.However in the earlier
work-Basic concepts a the definition of ring did not include an
identity.If it had one ,it was called a ring with an identity and the
one element ring was included (1=0).He had given examples and goes on
to say "all of the previous examples were were commutative with
identity ,An example of arising without an identity is the set of even
integers ". I see that in a commutative ring with 1 and no zero
divisors that any proper ideal is a subring with no identity. Can the
ideal have an identity if there are zero divisors ? I thought you would
all enjoy this bit of nostalgia.Good Night ,Stuart M Newberger. PS rngs
is very clever and ok,but to put a 1 in the definition of ring (to
which I am neutral;By the way S.Lang does this in his Algebra) and then
to speak of a ring without an identity is an oxymoron . and good luck
to David Bernier with Eisenbud.

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