Suppose is a finite ring with an odd number of invertible elements. Prove that has characteristic .
Assume that, for every , . Then, if we associate each element of with its opposite (which is in as well), we partition in pairs, contradicting the fact that the cardinality of is odd.
As a consequence, there exists such that . Multiplying by , this implies (where is the unit element of ), or (where ): the characteristic is .