A special class of commutative rings Can we characterize all commutative finite rings the sum of all elements is zero? For instance, all odd characteristic finite commutative rings have this property (easy). GF(2) does not.
 A: Since you're only using addition, you might as well ask this for abelian groups instead of rings.  The sum of all the elements of a finite abelian group is equal to the sum of elements of order $2$ (since the others cancel their inverses), which (together with $0$) form a subgroup which is a vector space over $\mathbb{F}_2$.  The sum of all elements of a finite-dimensional vector space $V$ over $\mathbb{F}_2$ is $0$ iff the dimension is not equal to $1$ (there are various ways you can see this; for instance, the sum must be invariant under all invertible linear maps $V\to V$, and we can map any nonzero element of $V$ to any other by an invertible linear map, so if the sum is nonzero there can only be one nonzero element).
So we conclude that the sum of the elements in a finite abelian group is nonzero iff it has exactly one element of order $2$, in which case the sum is that element of order $2$.  In terms of the classification of finite abelian groups, this means that when you write the group as a direct sum of cyclic groups, there is exactly one summand whose order is even.
