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+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?
> 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.