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.