```Date: Feb 4, 2013 12:19 AM
Author: Butch Malahide
Subject: Re: Finite Rings

On Feb 3, 11:09 pm, quasi <qu...@null.set> wrote:> William Elliot <ma...@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+xDoes this mean that the additive group of R is a group of order 7?> 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 RIf r is nonzero, then r is an element of order 2 in the additivegroup? What about Lagrange's theorem?>    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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