Date: Feb 6, 2013 8:33 AM
Author: quasi
Subject: Re: Finite Rings

William Elliot <marsh@panix.com> wrote:
>
>[In forum "Ask an Algebraist", user "Anu" asks] (edited):
>

>> If R is a finite commutative ring without multiplicative
>> identity such that every nonzero element is a zero divisor,
>> must there necessarily exist a nonzero element which
>> annihilates all elements of R?


Another attempt ...

Let Z_3 denote the ring of integers mod 3. and let R denote the
polynomial ring Z_3[x,y]. In the ring R, consider the ideals

M = (x,y)

I = (x^3 - x, y^3 - y, xy^2 - xy, yx^2 - xy)

Let T denote the ideal M/I in the quotient ring R/I.

Claim the ideal T qualifies as a ring without identity such
that, in the ring T, all nonzero elements are zero divisors,
and no nonzero element annihilates all elements of T.

An alternate way to interpet the above example is as follows.

Let T be the set of 3^5 expressions of the form

a*x + b*y + c*x^2 + d*xy + e*y^2

where a,b,c,d,e are in Z_3, represented as integers in the
range 0 to 2 inclusive.

Operations on T are the usual algebraic polynomial operations,
with coefficient operations mod 3, and the following additional
relations:

x^3 = x

y^3 = y

xy^2 = xy

yx^2 = xy

I'm not 100% sure my new example works, but I think it does. Of
course, I also thought my previous examples were valid, and they
weren't, but this one had the benefit of my understanding of
what was wrong with the previous ones.

quasi