Two questions:
1. If R is a ring then is every element of R either a unit or zero-divisor.
COnversely
2. IF R is a ring, then there is no elements of R which is both a unit and a zero-divisor.
Is R a finite ring?
Because, if it is then we can prove it as follows:
If x is a zero-divisor the proof is complete.
If x is not a zero-divisor then consider the finite set,
{a_1x,a_2x,...,a_nx} where a_i are the non-zero elements of the ring (assuming it is non-trivial).
Then, we can show none are equal to each other.
(Excercise).
Then by Dirichlet's Pigeonhole Principle, is an enumeration of all non-zero elements in R. Hence we can find an a_i such that a_i x = 1. Thus, x is a unit.
The Ring is not defined as a finite ring. These are true/false questions I have encountered in a abstract algebra class. If false I have to give a counterexample. For example, If R is a ring, then every element of R is either a unit or zero-divisor. In my mind this is false but the counterexample is what is messing with my head.