Date: Feb 4, 2013 2:38 AM
Author: quasi
Subject: Re: Finite Rings
quasi wrote:

>quasi wrote:

>>quasi wrote:

>>>Butch Malahide wrote:

>>>>quasi wrote:

>>>>> William Elliot wrote:

>>>>>

>>>>> >[In forum "Ask an Algebraist", user "Anu" asks:]

>>>>>

>>>>> >> If R is a finite commutative ring without multiplicative

>>>>> >> identity and if every element is a zero divisor, then does

>>>>> >> there exist a nonzero element which annihilates all elements

>>>>> >> of the ring?

>>>>>

>>>>> >No - the trivial ring.

>>>>> >So add the premise that R has a nonzero element.

>>>>>

>>>>> Even with that correction, the answer is still "no".

>>>>>

>>>>> Consider the commutative ring R consisting of the following

>>>>> seven distinct elements:

>>>>>

>>>>> 0, x, y, z, x+y, y+z, z+x

>>>>

>>>>Does this mean that the additive group of R is a group of order 7?

>>>>

>>>>> Besides the usual laws required for R to be a commutative

>>>>> ring (without identity), we also require the following

>>>>> relations:

>>>>>

>>>>> r^2 = r for all r in R

>>>>>

>>>>> r+r = 0 for all r in R

>>>>

>>>>If r is nonzero, then r is an element of order 2 in the additive

>>>>group? What about Lagrange's theorem?

>>>

>>>Oops. I missed the element x+y+z.

>>>

>>>But then x+y+z is an annihilator -- which destroys my attempted

>>>counteraxample.

>>

>>That's wrong too.

>>

>>x+y+z is not an annihilator.

>>

>>However it's not a zero-divisor -- in fact, it's an identity,

>>

>>Thus, my example still fails.

>>

>>>>> xy = yz = zx = 0

>>>>>

>>>>> Note that the above relations imply

>>>>>

>>>>> (x+y)z = (y+z)x = (z+x)y = 0

>>>>>

>>>>> so every element of R is a zerodivisor.

>>>>>

>>>>> However, since all elements of R are idempotent, it follows

>>>>> that no nonzero element of R annihilates all elements of R.

>

>Ok, I think I now have a valid counterexample.

>

>I'll post it shortly in a reply to the original post.

Forget it.

My new counterexample just vaporized.

At this point, I'm not so sure there there _is_ a

counterexample.

quasi