William Elliot <firstname.lastname@example.org> wrote: > >[In forum "Ask an Algebraist", user "Anu" asks] (edited): > >> If R is a finite commutative ring without multiplicative >> identity such that every nonzero element is a zero divisor, >> must there necessarily exist a nonzero element which >> annihilates all elements of R?
Ok, I think I have it now.
Consider a commutative ring R consisting of the following 8 distinct elements
0, x, y, z, x+y, y+z, z+x, x+y+z
obeying the usual laws required for R to be a commutative ring (without identity), and also satisfying the following conditions:
x^2 = x, y^2 = x, z^2 = x
r+r = 0 for all r in R
xy = yz = zx = 0
Then every nonzero element is a zero divisor but no nonzero element annihilates all elements of R.