Date: Feb 4, 2013 12:19 AM
Author: Butch Malahide
Subject: Re: Finite Rings

On Feb 3, 11:09 pm, quasi <qu...@null.set> wrote:
> William Elliot <ma...@panix.com> 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?

>    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.