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

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Posts: 11,309
Registered: 7/15/05
Re: Finite Rings
Posted: Feb 4, 2013 12:09 AM
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William Elliot <marsh@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

Besides the usual laws required for R to be a commutative
ring (without identity), we also require the following

r^2 = r for all r in R

r+r = 0 for all r in R

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.


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