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I'm not sure how to go about finding the solution to this question.

Let R be a ring with identity such that $x^2 = 1_R$ for all $0_R \neq x\in R$. How many elements are in $R$?

I've just been playing around with squaring elements, like $$(x+1_R)^2 = x^2+2.x+1_R =2.x+2.1_R = 1_R.$$

But I'm not sure where to go with this. Any help?

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    Either $1_R+1_R=0$ or $(1_R+1_R)^2=1_R$ so $3\cdot 1_R=0$.2012-03-05

2 Answers 2