Re: A question in noetherian ring!

On Sun, 29 Jul 2007 21:16:50 -0700, Cooper <cooper0040@xxxxxxxxx>
wrote:

On 7 30 , 11 14 , quasi <qu...@xxxxxxxx> wrote:
On Sun, 29 Jul 2007 09:33:04 -0700, Cooper <cooper0...@xxxxxxxxx>
wrote:

Hi!

The question I want to ask is following;

Let M be a finitely generated R-module, where R is noetherian. Suppose
I is an ideal of R such that for each element a in I, there exists a
nonzero element x in M s.t. xa=0. Show that xI=0 for some nonzero
element x in M.

Initially, I was almost sure the above claim was false, but after
several failed attempts to produce a counterexample, I'm no longer so
certain. Perhaps it's true, but I remain skeptical.

Was this an exercise from a text? If so, which?

quasi

Yes. It is an exercise problem contained in chap 27 of Issac Martin's
Algebra text(of course graduate course)

Ok, in that case, I'll stop trying to disprove it.

Can I assume that R is a commutative ring with 1?

But R is not necessarily an integral domain, right?

Also, as a minor correction, xI should probably be written as Ix or
I*x (the ring elements multiply on the left).

quasi
.

Relevant Pages

  • Re: Prime numbers within Galois field?
    ... Suppose you're working _within_ a Galois Field, ... But in any field every nonzero element is divisible by every other ... are no irreducible elements. ...
    (sci.math)
  • Re: A question in noetherian ring!
    ... nonzero element x in M s.t. xa=0. ... It is an exercise problem contained in chap 27 of Issac Martin's ... Algebra text(of course graduate course) ...
    (sci.math)