```Date: Feb 4, 2013 12:33 AM
Author: quasi
Subject: Re: Finite Rings

Butch Malahide wrote:>quasi wrote:>> William Elliot 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>>Does 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 R>>If r is nonzero, then r is an element of order 2 in the additive>group? What about Lagrange's theorem?Oops. I missed the element x+y+z.But then x+y+z is an annihilator -- which destroys my attemptedcounteraxample.>>    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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