Suppose is a finite ring with an odd number of invertible elements. Prove that has characteristic .

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- Oct 28th 2008, 08:06 AMpetterfinite ring with an odd number of invertible elements
Suppose is a finite ring with an odd number of invertible elements. Prove that has characteristic .

- Oct 28th 2008, 10:36 AMLaurent
I denote by the set of invertible elements of .

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 .