Date: Feb 4, 2013 12:33 AM
Subject: Re: Finite Rings
Butch Malahide 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
>> 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 attempted
>> 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.