In a unital ring, (-1)² = 1

Let R be a ring with 1. Show that (-1)^2 = 1 in R.


Let x \in R. Then (-1)^2x = (-1)(-x) = x. Similarly, x(-1)^2 = (-x)(-1) = x. By the uniqueness of 1, (-1)^2 = 1.

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