Date: Feb 5, 2013 4:12 AM
Author: William Elliot
Subject: Re: Finite Rings

On Sun, 3 Feb 2013, William Elliot wrote:

> > 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?


Assume without debate nor dispute, that R has a non-zero element p.

If there's no nonzero annihilator, then for all nonzero x,
there's some a_x with nonzero a a_x.

Let p0 = p and uning induction, for all j in N, define
p_(j+1) = pj a_pj.
p0, p1, p2, ... is sequence of nonzero elements.

Since R is finite, there's some distinct j,k with pj = pk.
Thusly we've some nonzero a,b with ab = a. Now what?