Re: Coprimeness - I think I'm confused, but I'm not sure

David Rosoff wrote: 
On Fri, 1 Apr 2005, Matt Gutting wrote:

Arturo Magidin wrote:

In article <1112387619.8214cb7fca92c72ca9857edb4554270a@teranews>,
Matt Gutting  <matthewdba@xxxxxxxxxxx> wrote:

In light of some of the comments in the JSH threads involving coprimeness and Z[1/2]:

If the definition of coprimeness is something like:

To say that 'p and q are coprime in a ring' is to say
'there exist a,b in the ring with ap + bq = 1'




This is true in rings with 1.



Okey dokey. I guess it's difficult for me to work with rings
lacking a unit element because the books I've used (Herstein,
and Rotman), while mentioning that "not all mathematicians
insist that a ring contain a unit element", do in fact insist
for the purposes of the book that rings be required to have
a unit element.

You wouldn't know of a (beginning grad-level) book that discussed
the differences between rings with and without unit element?


Check out Harry C. Hutchins' book, "Examples of Commutative Rings." There is a shortish section about rings without 1. I agree with Dale Hall's sentiment, expressed in one of the JSH threads, that not having a 1 makes ideals turn stupid. For example, in such a ring, maximal ideals need not be prime. Let R = 2Z, M = 4Z. Then M is a maximal ideal, but it is not prime (2*2 is in M).

The book is also quite good for one who wants more familiarity with the menagerie of commutative rings, especially the differences between Noetherian and non-Noetherian rings. I particularly liked the example of a ring R of Krull dimension 1 such that R[x] has Krull dimension 3. Unfortunately I don't remember the construction.

HTH,

-dave

Thanks Dave!

My reference book on these subjects is Joseph Rotman, "Advanced Modern
Algebra" (no edition given; I'm assuming it's first edition). Do you or
anyone else reading know anything about how good it's considered? I
can't find a review on the AMS site archives. (But then I don't know
whether they generally review introductory texts like this.)

Matt
.

Relevant Pages

  • Re: Coprimeness - I think Im confused, but Im not sure
    ... lacking a unit element because the books I've used (Herstein, ... the differences between rings with and without unit element? ... Or, more strongly, are any 2 rationals coprime in Q? ...
    (sci.math)
  • Re: Coprimeness - I think Im confused, but Im not sure
    ... lacking a unit element because the books I've used (Herstein, ... the differences between rings with and without unit element? ... I agree with Dale Hall's sentiment, expressed in one of the JSH threads, that not having a 1 makes ideals turn stupid. ... I particularly liked the example of a ring R of Krull dimension 1 such that Rhas Krull dimension 3. ...
    (sci.math)