Re: Question about zero divisors in finite rings

On Wed, 12 Sep 2007 17:26:03 -0700, Rotwang <sg552@xxxxxxxxxxxxx>
wrote:

Brian VanPelt wrote:

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 don't think that this works in general, since k*1 might be 0, for
example. This is the case when the ring in question is the field with
four elements that Arturo gave, with k=2.

Exactly, but it does work starting with 6, the smallest natural number
that is the product of two distinct primes.

This brings up a funny story. In a lot of algebra books they mention
fields were 2 does not equal zero, and I love to show that to nonmath
people who look at me with total confusion.

Brian

.

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