Re: rings with finitely many principal ideals

On Sun, 02 Sep 2007 20:59:09 -0400, quasi <quasi@xxxxxxxx> wrote:

On Sun, 02 Sep 2007 22:18:19 +0200, Jannick Asmus
<jannick.news@xxxxxx> wrote:

On 02.09.2007 22:08, Jack Schmidt wrote:
On Sep 2, 3:35 pm, quasi <qu...@xxxxxxxx> wrote:
Does there exist a ring R with only finitely many principal ideals but
infinitely many ideals?

quasi

An ideal is the sum of the principal ideals contained within it.

Thus a ring with finitely many principal ideals is both Noetherian and
Artinian, hence the ring is 0-dimensional and is the finite product of
local Artinian rings.

So must a local Artinian ring with finitely many principal ideals have
finitely many ideals?

quasi

Whoah, I don't think locality is important for this answer. Let's go
back to modules over a ring for a second. A ring R is (left, right or
both) Aritinian if it has the same property as a (left, right or both)
R-module.

Now, an R-module is Aritnian if it obeys the Descending Chain
Condition - every descending chain of submodules stabilizes. Ideals
are submodules of the ring viewed as a module over itself. So, the
language of modules is appropriate here with the added benefit of
being more general.

Let A be a left, Artinian module over a ring R, and suppose A_1, A_2,
.... are a collection of submodules - I'm not saying they are all
dstinct from each other because that would be giving too much away.

Then, what do you make of

A_1 intersect A_2

A_1 intersect A_2 intersect A_3

A_1 intersect A_2 intersect A_3 intersect A_4

etc?

If you wan't, substitute the word ideal for module everywhere to
achieve your desired result.

Thanks,

Brian






.

Relevant Pages

  • Re: real valued modulo math
    ... link you site does actually list a continuum ... A field F has only trivial ideals, ... the ring and consider the elements of the ring from ... though again in the usage of the quotient ring I do ...
    (sci.math)
  • Re: real valued modulo math
    ... the ideals generated by 2.1 or 2.3, ... the ring and consider the elements of the ring from ... though again in the usage of the quotient ring I do ... In topology, we define the quotient of a space ...
    (sci.math)
  • Re: -- Maximal ideals in Z[x] and Z[x, y]
    ... THEOREM TFAE for a semigroup ring R, with unitary ring R, ... i.e. a ring where every ideal is a finite product of prime ideals. ...
    (sci.math)
  • Re: real valued modulo math
    ... the ring and consider the elements of the ring ... that are not trivial, i.e., ideals that are not ... though again in the usage of the quotient ring I ... In topology, we define the quotient of a space ...
    (sci.math)
  • Re: JSH: The "Published" paper he dosent what you to know about.
    ... Yet you don't even know what a ring is, ... You imagine that the situation of 2 & 6 being coprime in the ring ... which ideals are prime, and how do those ideals respect ... integer functions when t is an algebraic integer. ...
    (sci.math)