Re: Question about zero divisors in finite rings

On Wed, 12 Sep 2007 04:55:22 GMT, Gerry Myerson
<gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <kuiee39s3pni8ah478mk7fcqv0tt9rejdb@xxxxxxx>,
Brian VanPelt <brvanpelt@xxxxxxxxxxxxx> wrote:

On Tue, 11 Sep 2007 18:07:54 -0700, Rotwang <sg552@xxxxxxxxxxxxx>
wrote:

While reading an old thread the other day I saw an article in which a
poster asserted that any ring with 6 elements necessarily contains
zero divisors. Having thought a bit about why this is true I think I
can show that a ring with n elements must have zero divisors provided

i) n is composite, and
ii) no prime appears in the prime factorisation of n with exponent
greater than one.

My question is: is this correct, and is it a special case of a more
general fact?

If n is composite with unity, then i) is all you need. If n = km,
then the k*1 times m*1 is the ring zero.

I don't know what it means for a number to be "composite with unity,"
but the field of 4 elements is a ring with a composite number of
elements and no zero divisors.

Sorry, I don't know what it means either and I wrote it!

What I meant was to suppose that R was a ring with unity, and n was
composite. Say the unity is called 1. Then k*1 represents 1 added to
itself k times. If n = km, and * is the ring multiplication, then

k*1 * m*1 = (km)*1

which would be zero in this ring.

I didn't see how to do this without a unity element and I honestly
don't know it is true.

It looks like the poster wants to know that is i) and ii) are both
true, with no other restrictions on the ring, then the ring will
contain zero divisors. Now, I haven't read all the responses yet to
know if a counter-example was exhibited, but I did want to clarify
what I said.

Sorry,

Brian

.

Relevant Pages

  • Re: An infinite number question
    ... If you exclude 0, the remaining numbers form a group ... A set with both addition and multiplication is a ring, ... If the set is a cyclic group under addition, as would be the infinite ... division except by zero is possible. ...
    (talk.origins)
  • Re: Can you explain prime force/energy & matter?
    ... >>a photon, because infinite energy is not available. ... Zero is zero. ... step up to spinning your ring (like ... So if we continue to hold the precession at ...
    (sci.physics)
  • Re: Gravity Inside A Cylinder
    ... Binney and Tremaine I find assuming a zero thickness ring ... The net force in the system is zero when the sun is located ... for all positions within the ring plane except the center. ...
    (sci.physics)
  • Re: Do dark matter calculations make the stupid assumption that all mass is at the center of the gal
    ... Gravitational Field from a Ring of Mass ... plane of the ring. ... If x is zero such that the particle is within the ... napkin calculation showing that dark matter actually exists. ...
    (sci.physics)
  • Re: An uncountable countable set
    ... think the absolute minimal set of axioms: ... semigroup under multiplication, distributive law), you have (for any x, ... y, in the ring): ... unless it is simply zero. ...
    (sci.math)