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Topic: Finite Rings
Replies: 28   Last Post: Feb 6, 2013 8:33 AM

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quasi

Posts: 10,450
Registered: 7/15/05
Re: Finite Rings
Posted: Feb 4, 2013 8:56 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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?


Ok, I think I have it now.

Consider a commutative ring R consisting of the following
8 distinct elements

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

obeying the usual laws required for R to be a commutative
ring (without identity), and also satisfying the following
conditions:

x^2 = x, y^2 = x, z^2 = x

r+r = 0 for all r in R

xy = yz = zx = 0

Then every nonzero element is a zero divisor but no nonzero
element annihilates all elements of R.

quasi



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