quasi
Posts:
9,080
Registered:
7/15/05
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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
relations:
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.
quasi
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